Amenability of Certain Banach Algebras

Home , Amenable Banach algebra

CHENG Yin Hei

A Thesis Submitted in Partial Fulfilment

of the Requirements for the Degree of

Master of Philosophy

Mathematics

�Th eChinese University of Hong Kong

June 2006

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. ！ Professor Ng Kung Fu (Chair) Professor Leung Chi Wai (Thesis Supervisor) Professor Luk Hing Sun (Committee Member) Professor Huang Li Ren (External Examiner) Amenability of Certain Ba.na.ch Algebras i Abstract

The notion of amenability, is an important one in abstract harmonic analysis. B.E.

Johnson showed that the amenability of a locally compact group can be characterized in terms of the Hochschild cohomology of its group algebra, thus initiated the theory of amenable Banach algebras. This survey is mainly focused on the problems relating the amenability of locally compact groups and the Banach algebras associated with the given groups. We will also see how the properties of these algebras and the relationships between them in the commutative case are generalized to the non-commutative case. 摘要

馴服性在抽象調和分析中是一個重要的槪念。B.E. Johnson證明了局部緊緻群的馴服性是隱藏於其群代數的Hoschild上同調中，這結果後來開創了馴服巴拿赫代數的理論。本文的重點在於局部緊緻群及其相關巴拿赫代數的馴服性問題，當中也會涉及到這些代數的性質和其相互關係如何從交換情形推廣至非交換的一般情形。 Amenability of Certain Ba.na.ch Algebras ii Introduction

A locally compact group G is said to be amenable if there is a left invariant mean on

L°°{G). If the axiom of choice is assumed, then we have the following strange phenomenon:

"An orange can be cut into finitely many pieces, and these pieces can be reassembled to

yield two oranges of the same size as the original one."

This is in fact an application of the so-called "paradoxical decompositions". One of the

key ingredients for the proof of this paradox was that F2, the free group of two generators,

lacks the property of amenability.

In the theory of abstract harmonic analysis, the group algebra L^{G) is undoubtedly

the most important Banach algebra. It is thus very natural to ask how to use the group

algebras to characterize the amenability of the given groups. In 1972，B.E. Johnson

answered this question completely. He showed that G is amenable if and only if the first

Hoschild cohomology group H{V(G), E*) = {0} for any dual Banach L^(G)-bimodule

E. This condition makes sense for arbitrary Banach algebras, which may have nothing

to do with locally compact groups. Consequently, it is meaningfully to define "amenable

Banach algebras".

On the other hand, Eymard found that the Fourier algebra A{G) is the “ dual" object of the group algerba If G is abelian, A{G) is nothing but the image of the

Fourier transform of The name of the theory of ^4(6') is therefore suggested as

“non-commutative abstract harmonic analysis，’. As a Banach algebra, the most important property of the group algebra should be the existence of the bounded approximate identity.

Many nice results on L^(G) in fact depend on this fact. The Fourier algebra, which is the dual object of the group algebra, unfortunately does not own a. bounded approximate identity in general. In 1968, Leptin proved that A{G) has a bounded approximate identity if and only if G is amenable. In the above point of view> amenable property becomes a reasonable assumption in the study of A{G). Amenability of Certain Ba.na.ch Algebras iii

This survey article is divided into four chapters. All the basic results in the general

theory of amenable Banach algebra will be presented in Chapter 1. The. general properties

of group algebras and measure algebras will be founded in Chapter 2. We will also see how

the conditions on them characterize the amenability of a given group. As said above, the

amenability assumption is a suitable one in the study of A{G) when considering the "dual

version" of the properties of L^(G). We will see that these “dual" properties are in fact

equivalent to the amenable assumption on G. Another interesting question is to capture

the amenability of a locally compact group G through the different sort of amenability of

A{G), which is called "operator amenability". In this case, the operator space structure of

A{G) is taken into account. Other operator cohoinological properties are also considered in this chapter. Amenability of Certain Ba.na.ch Algebras iv

ACKNOWLEDGMENTS

I am deeply indebted to my supervisor, Prof. Chi-Wai Leung, not only for his im- measurable guidance but also his kind encouragement and industrious supervision in the course of this research programme. To gain techniques and insights under his supervision is undoubtedly my lifetime benefit. I am also very grateful to Prof. Anthony To-Ming

Lau, Prof. Marty Walter and Prof. Nico Sprorik. They were willing to explain my ques- tions patiently in the preparation of this thesis. Moreover, I would like to acknowledge my classmates, Mr Yiu-Keung Poon and Mr Wai-Kit Chan, for their help and discussion during the program. I particularly would like to thank Mr Pak-Keimg Chan for finding a series of papers for me. Finally, I wish to thank my parents and Ms Alice Law, for giving me their constant love, encouragement and support. Contents

1 Preliminaries 1

1.1 Haar measures, Group algebras and measure algebras of lo-

cally compact groups 1

1.2 Banach algebras and amenability 7

2 Cohomological properties of Group algebras and Measure

algebras 13

2.1 Cohomological properties of L^{G) 13

2.2 Amenability and weak amenability of M{G) 16

3 Amenability and Fourier algebras with Bounded Approxi-

mate identities 20

3.1 Properties of Fourier-Stieltjes algebras, Fourier algebras, group

von Neumann algebras 21

3.2 Conditions on A{G) that characterizing the amenability of G 25

4 Operator Amenability and Fourier algebras 45

4.1 Operator Amenability 45

V Amenability of Certain Ba.na.ch Algebras vi

4.2 Hopf-von Neumann algebras and their predual structures . 50

4.3 Operator Cohomological properties of A{G) 51

4.4 Ideals in A{G) with bounded approximate identities .... 60

4.5 Other Cohomological properties of A{G) 64

A Objects or notions and their "dual" versions 69

B Results and "dual results" for general locally compact groups 70

C Results and "dual results" which are equivalent to the

amenability of the group 71

Bibliography 72 Chapter 1

Preliminaries

We shall have some basic results in this chapter which will be used later.

1.1 Haar measures, Group algebras and measure al-

gebras of locally compact groups

The sources of this chapter are [Fol] [Chapter 2,3], [Run] [Appendix A] and [Dal] [Chapter

3] unless specified.

Definition 1.1.1. Let G be a locally compact group. A Haar measure on G is a non-zero

Borel regular measure ^g on G such that /^g is translation invariant:

(ie) ijlg{xE) = fJ^ciE) for any Borel set in G

Theorem 1.1.2. Let G be a locally compact group. Then there exists a Haar measure fic on G. Moreover, Haar measure on a locally compact group G exists uniquely up to a positive multiple.

Proposition 1.1.3. For a locally compact group G with a Haar measure fiQ, let U be an open subset in G, and let K be a compact subset in G. Then Hg{U) > 0 and fJ^ci^^) < oo.

Theorem 1.1.4. Let G be a locally compact group with a Haar measure jic. Then we have:

1 Anieiiability of Certain Banach Algebras 2

(a) G is compact if and only if ilig{G) < oc

(b) G is discrete if and only if iiq is a positive multiple of counting measure.

Definition 1.1.5. Let G be a locally compact group with a Haar measure iig- If for

any a: G G, we define /J^xci^) = lM3�E:c) the, n /-ixci^) is again a left Haar measure. By

the uniqueness of Haar measures, there is a function A (a:) such that ii^g —从工)and

A(x) is independent of the oringinal choice of fiQ. The function A : G—— >(0 ’ oo) thus

defined is called the modular function of G. Furthermore, G is said to be unimodular if

A三1.

Theorem 1.1.6. Let G be a locally compact group. If G is compact or abelian, then G

is unimodular.

Theorem 1.1.7. Let G be a locally compact group with a Haar measure Then

(a) A is continuous homomorphism from G to Ex-

(b) For any f eL\iJic),

[f{xy)dfiGix) = A{y-') [ f(x)d^a{^) JG JG

f fix-')A{x-'')dmGix) = [ f{x)dmG{x) JG JG

From now on, any group will be assumed to be locally compact, and G will denote a locally compact group. For any locally compact group, we fix a Haar measure mo on G.

For any compact group G, we choose the Haar measure rric on G such that mc{G) 二 1.

For any discrete group G, we choose ttig to be the counting measure.

Definition 1.1.8. Let 1 < p < oo and let be the set of all p-integrable function on G with respect to mo. Let /i,/2 G fi and /之 are said to be equivalent if

II/i — f2\\p = 0. Write the set of all equivalent classes in Anieiiability of Certain Banach Algebras 3

From now on, we use the following notation without further specification. Denote by

• Cb{G) the space of bounded continuous functions on G.

參（7o(G) the space of continuous functions vanishing at infinity on G.

• Cc(G) the space of continuous functions with compact support on G.

Proposition 1.1.9. Ifl

is II • lloo" dense in Co(G)‘

Definition 1.1.10. Let M{G) be the set of all complex Radon measure on G. Define a

norm || . ||m(g) by

IImIIm(g) ：= M{G) ill e M{G))

Also, define the convolution operation * on M(G) by

f f{x)d{fz*iy)(x) := I {[ f{xy)dfi{x))diyix) [f E e M{G)) JG JG JG Definition 1.1.11. A measure in M{G) is said to be discrete if it has the form fi —

J2S£G where f G An element /j, G M{G) is called a continuous measure if

MW) -0(56 GO.

Theorem 1.1.12. The following statements are true for all locally compact group G:

(a) (M(G'), II • ||M(G), *) is a unital Banach *-algebra and the unit is given by the point

mass measure at identity

(b) The set Md{G) of discrete measures in M{G) is a closed, unital *-subalgebra of

M(G) identified with = M{G) if and only if G is discrete.

(d) The set Ma{G) of measures in M{G) absolutely continuous with respect to the Haar

measure is a closed *-ideal in M{G) identified with L^{G) via

iME) := / f{x)dmG{x) JE Anieiiability of Certain Banach Algebras 4

(e) The convolution operation * on inherited from M{G) is given by

f * g{y) = / f(yx)g{x~'^)dmG{x) JG

(f) The set Mcs{G), a subset of Mc{G), consisting of the continuous measures which

are singular with respect to the Haar measure is a closed sub space of M(G) and

Mc{G) == Ma{G) e Mcs{G); Ma{G) = M(G) if and only if G is discrete.

(g) M(G) = Ma(G) © Mcs(G) ©1 Md(G) and suppose (jl e M{G) has the decomposition

= /^a + /J'cs + fJ'd, then its total variation has the decomposition \/j,\ = |/ia| + |/“| +

Definition 1.1.13. Let X be a locally compact Hausdorff space X with a Radon measure

fj.. A set 丑 C AMs locally Borel if A F is Borel whenever F is Borel and iJ,{F) < oo.

A statement about points in X is true locally almost everywhere if it is true expect on a locally null set. A function f : X ——> C is locally measurable if f~^(A) is locally Borel for every Borel set A C C. Define L°°(/i) to be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locall}^ a.e.

is a Banach space with norm

ll/lloo ：= {c ： I/I < c locally a.e.}

Remark. We have (Lp(G))* ^ L'^{G) if i + i - 1 and 1< p, g < oo, (L\G)y ^ L�[G) and (CoiG))* = M{G) from simple Banach space theory.

Theorem 1.1.14. Suppose I

f * 9{y) = / fiyx)g{x-'^)dmG{x) JG exists a.e. and we have f * g e LP[G) and \\f * < |I/||I|MIP. If p = OO, then f * g e C“G).

Definition 1.1.15. Let 1 < p < oo, ^ G M(G) and f e If{G). Define the function /z*^/ by

("*p/)�：= / f(y~^x)di.L(y) Jc Anieiiability of Certain Banach Algebras 5

If / G define the function f * by

：二 [ f{xy--')A[y-')df,{y) JG

If / G L°°{G), define the function / * by

:= / f{xy-^)dfi{y) JG

Remark. By definition 1.1.15, if 1 < p < oo, /i G M{G) and f e then L工-if =

4 * /； If / e L'{G), then A(x)RJ = f * if e L\G).

Theorem 1.1.16. Suppose fi G M(G), g G L^{G). Then the function // * / and / * // exists a.e. and we have A^* /，/*". G L\G) and \\f * mI|i，IIm * /||i < ||"||m(g)||/||i.

Thus, (L\G), II • 111, *) is a closed ideal of (M(G), || . ||M(G)’ *).

Theorem 1.1.17. //1 < p < oo and f e Lp{G), then

\\Lyf - /lip, WRyf — /lip 0 as y^ec

Let / be a function on G. We say that f is left(resp. right) uniformly continuous if

\\Lyf — f\\oo 一 0(resp. \\Ryf — /||oo 0) as y e(;. If f is both left and right uniformly continuous, it is said to be uniformly continuous. Denote by LUC(G) the space of left uniformly continuous functions on G, RUC{G) the space of right uniformly continuous functions on G and UCB{G) the space of uniformly continuous functions on G.

Theorem 1.1.18. Let B be a the set of all open symmetric neighborhoods of cq• Define a partial order on B by

Ui ^ ih if U2 C Ui

(a) is an approximate identity of which is self-adjoint with each

term having L^ — norm, equal to 1. Anieiiability of Certain Banach Algebras 6

(b) If 1 < ]) < oo, then we have

赤小命)‘丨丨.丨丨“

(d) Iffe LUC{G), then

maiU) J j

(e) {„,J(U)Xu}ueB is a hounded approximate identity of M(G) in the weak sense:

问 ^^Su) * "，" * ；^Ty 一'几•(外腳)）（",e 專)）

Thus, L^(G) is cr{M{G),Co{G))-dense in M{G)

Theorem 1.1.19. Let G be a locally compact group. Then we have: .

(a) LUC(G) = L乂G) * L1G)

(b) RUCiG) = L°°{G) * L\G)

Theorem 1.1.20. Let G, H be locally compact groups. Then

L\G)%L\H) ^ L\G X H)

Theorem 1.1.21 (Wendel's theorem). Let T be a left, multiplier acting on L\G).

Then there exists a unique complex regular measure u on G such that T{F') — / * for all f belonging to

Proof. Refer to [Wen] [Theorem 1]. •

Theorem 1.1.22. T. J� ={T e m{L\G),L\G)) : ||T(/)||i - ||/lh for any f e L'(G)}

Proof. Refer to [Wen] [Theorem 3]. • Anieiiability of Certain Banach Algebras 7

Theorem 1.1.23. Let T : L^(Gi)——> L^{G2) he a norm-decreasing isomorphism. Then

there exists a unique isomorphism t : Gi—— >G2 and a continuous character x : —> C

such that

(a) TRxT~^ = for all x belonging to G

(b) {Tf)(r{x)) — CT(x)f{x) for all x belonging to G

where mcAE)

饥G “丁则 for some Borel set E. Hence T is actually an isometry.

Proof. Refer to [Wen] [Theorem 5]. •

Corollary 1.1.24. Let G], G2 be locally compact groups. Then Gi and G2 are topologi-

cally isomorphic if and only if I八Gi) and L^{G2) are isometrically isomorphic.

Theorem 1.1.25 (Johnson's theorem). Let G、，G'2 be locally compact groups. Then

Gi and G2 are topologically isomorphic if and only if M(Gi) and M{G2) are isometrically

isomorphic.

Proof. Refer to [Johl] [Corollary of Theorem 3]. •

1.2 Banach algebras and amenability

All of the results in the following can be founded in [Run] [Chapter 2].

Definition 1.2.1. Let ^ be a Banach algebra and let E be a Banach space.

(a) A left(resp. right) ^-module E is called a. left(resp. right) Banach A-module if there

is K > 0 such that

||a.:c|| S/c||a||||:r|| (aeA^e 约

{resp. < fc!|a||||a:|| {aeA.xeE)) Anieiiability of Certain Banach Algebras 8

(b) An ^-^-bimodule E is called a Banach A-B-bimodule if it is both a left Banach

y4-niocliile and right Banach 5-modiile.

and right Banach ^-module.

Definition 1.2.2. Let E, F be Banach ^-bimodules. A map $ : E——> F.

^(u • x) •=U- {u e A,x e E)

is said be a bounded left multiplier of E into F. Let QJlyi {E^ F) be the space of all bounded

left multipliers of E into F. If ^ is a Banach algebra, A becomes a Banach A-bimodule

in a. natural manner. We simply write 97l‘4(A A) := A).

Definition 1.2.3. Let X be a Banach algebra, and let E he £i Banach /1-bimodule.

(a) A bounded linear mapping D : A > E is called a derivation if

D(ah) = a • D(b) 4- D(a) • b (a, be'A)

The set of all derivations from A into E is denoted by E).

(b) Let X e E. We define ad^ : A ^ E,a ^ a • x — x • a. It is easily seen that

adx G E). Derivations of this form are called inner derivations. The set of

all inner derivations is denoted by E).

cohomology group of A with coefficients in E.

Definition 1.2.4. Let ^ be a Banach algebra. If is a Banach ^-bimodiile, then it is easily seen that E* becomes a Banach yl-bimodiile through

< x,(f)' a >•=< a • a;,(/> >, < x, a • 0 >:=< x • a,0> (a e A,x e E,(f) e E*)

The dual spaces of Banach bimodiiles equipped with these module operations are called dual Banach bimodules. Anieiiability of Certain Banach Algebras 9

Proposition 1.2.5. Let A he a Banach algebra，and let E, F be A-bimodules. Then

E ® F becomes an A-bimodule under the actions:

a • (x

Proposition 1.2.6. Let A be a Banach algebra, and let I be its dosed ideal. Then All

becomes a Banach A-bimodule through

6 • (a + /) = 6a -f / and (a + I) . b = ab + I (a, b e A)

Definition 1.2.7. Let A be a Banach algebra.

(a) If v4 is a unital Banach algebra, a Banach i4-bimodule E is said to be united if there

exists ca ^ A so that ca - x = x • ca = x (x G E)

(b) A Banach ^-bimoclule E is said to be pseudo-unital ii E = A • E • A.

(d) A Banach y4-bimodule E is said to be essential if span{A • E • ^4)'' = E.

Definition 1.2.8. Let Ehea Banach ^-bimodule. A left(resp. right) approximate identity in A for X is a net {e^Jc^e/i in A such that

\\ea-x-x\\E 0 (.T e E)

{resp. • x - 0 (x e E))

,and is said to be hounded if supcieA\\&a\\A < oo.

An approximate identity is both a left and a right approximate identity.

Theorem 1.2.9 (Cohen's factorization theorem). If A has a hounded left (right) approximate identity for X, then X = A • X {resp. X = X • A). In particular, if A has a bounded approximate identity for X, then X 二 A • - A. Anieiiability of Certain Banach Algebras 10

Corollary 1.2.10. Let A be a Banach algebra with a bounded approximate identity, and

let X be a Banach A-bimodule. Then A • X, X • A and A - X • A are closed suhmodules of

Proposition 1.2.11. For a Banach algebra A with a bounded approximate identity, the

followings are equivalent:

(a) H^iA, E*) = {0} for any Banach A-bimodule E.

(b) H^ {A, E*) = {0} for any pseudo-unital Banach A-bimodule E.

Definition 1.2.12. If a Banach algebra A is contained as a closed ideal in another Banach

algebra B, then the strict topology on B with respect to A is defined through a family of

semi-norms (pa)a£A where

Pa(b) := \\ba\\ + \\ah\\ (b e B).

The corresponding topology is denoted by s{B,A).

Proposition 1.2.13. Let A be a Banach algebra with a bounded approximate identity

which is contained as a closed ideal in another Banach algebra B. Let E be a pseudo-

unital Banach A-bimodule, and let D e E*). Then E is a Banach B-bimodule

in a canonical fashion. Moreover, D has a unique extension D G E*) which is

continuous with respect to the strict topology {s(B, yl)) on B and the w*-topology on E*.

Definition 1.2.14. A Banach algebra A is called amenable if 二 {0} for any

Banach A-bimociiile E.

Proposition 1.2.15. Let A be an amenable Banach algebra. Then A has a bounded approximate identity.

Proposition 1.2.16. Let A be a Banach algebra, and let E, F be A-bimodules. Then

E ® F becomes a Banach A-bimodule through

a . (x 0 y) 二 [a • X� 0 y and {x ^ y) • a = x 0 {y • a) {a e A, x E E, y e F) Anieiiability of Certain Banach Algebras 11

From now on, 0 denotes the projective tensor product of Banach spaces.

Definition 1.2.17. If. A is a Banach algebra, then the corresponding diagonal operator

is defined through

A^ : A§>A ~~> A,a

Proposition 1.2.18. Let A be a Banach algebra. A®A becomes a Banach A-bimodule

through

a - (h^c) := ah 0 c and (b®c)-a:=h^ ca {a, b, c e A)

Then A^ is a bimoule homomorphism with respect to this module structure.

Definition 1.2.19. Let ^ be a Banach algebra.

(a) An element M G {A(S>A)** is called a virtual diagonal for A if

a - M =-- M -a and a • A^M = a (a e A)

(b) A bounded net {m^J j^^^P^ C (A~~A) is called an approximate diagonal for A if~~

~~a . rua — rrict • a —> 0 and aAA'rria —^ 0 (a G A)~~

~~Theorem 1.2.20. For a Banach algebra A, the followings are equivalent:~~

~~(a) A is amenable.~~

~~(b) There is a virtual diagonal for A.~~

~~(c) There is an approxiomte diagonal for A.~~

~~Proposition 1.2.21. Let A be an amenable Banach algebra, let B be a Banach algebra,~~

~~and let 9 : A —> B be a continuous homomorphism with dense range. Then B is~~

~~amenable.~~

~~Corollary 1.2.22. If A is an amenable Banach algebra and if I is a closed ideal in A, then A/I is amenable. Anieiiability of Certain Banach Algebras 12~~

~~Definition 1.2.23. Let E be a Banach space. A closed subspace F of E is said to be weakly complemented in E if F丄 is complemented in~~

~~Theorem 1.2.24. Let A be an amenable Banach algebra and let I be a dosed ideal of A.~~

~~Then the fallowings are equivalent:~~

~~(a) I is amenable.~~

~~(b) I has a hounded approximate identity.~~

~~(c) I is weakly complemented.~~

~~I Chapter 2~~

~~Cohomological properties of Group algebras and Measure algebras~~

~~2.1 Cohomological properties of L^{G)~~

In this section, we shall see that the amenability of a group can be characterized by the vanishing of certain cohomology group of V-(G). Thus, we may use the cohomological triviality condition to define the amenability of general Banach algebras. We shall also characterize an amenable group G by some properties of the ideals of Again, all of the proofs of the following results can be founded in [Run] [Chapter 2] unless specified.

~~Definition 2.1.1. Let G be a locally compact group. Let 丑 be a subspace of L°°(G).~~

~~Suppose that E contains the constant functions and is closed under complex conjugation.~~

~~(a) E is said to be left invariant (reps, right invariant) \i 5g^ (j) e E (resp. e E)~~

~~for ^\\(j)e E and g G G.~~

~~(b) A mean on E is a functional m e E* such that m(l) = ||m|| = 1.~~

~~(c) If E is left invariant(resp. right invariant), then a mean m on E is said to be left~~

~~invariant (resp. right invariant) if~~

~~< * 0’m�=< 0’m � (g e G,(f) e E)~~

~~13 Anieiiability of Certain Banach Algebras 14~~

~~(resp. < (f)^6g,m >=< 4>, m > {g 6 G, 4> e E))~~

~~(d) G is said to be amenable if there is a left invariant mean on L�(G).~~

~~Theorem 2.1.2. Compact groups and locally compact abelian groups are amenable.~~

~~Definition 2.1.3. Let G be a locally compact group.~~

~~(a) Let 丨丨=1 = {f e L\G) ： / < 0 and ||/||i = 1}.~~

~~(b) Let E be a siibspace of L°°(G) such that * E C E. A mean m e E* is~~

~~said to be topologically left invariant if~~

~~< / * 0，m�= <0，m� ((PeEJ eL'~~

Definition 2.1.4. Let G be a locally compact group. If tti and 兀2 are two unitary representations of G. We say that tti is weakly contained in 兀2 if /^er(7rf*) C Ker{7Tf*) where tt^" denotes the canonical representation on €"{0) corresponding to tt.

~~Theorem 2.1.5. Let G be a locally compact group Then the following statements are equivalent.~~

~~(a) G is amenable.~~

~~(b) There is a left invariant mean on Cb{G).~~

~~(c) There is a left invariant mean on LUC{G).~~

~~(d) There is a left invariant mean on RUC{G).~~

~~(e) There is a left invariant mean on UC{G).~~

~~(f) There is a topologically left invariant mean on UC(G).~~

~~(g) There is a topologically left： invariant mean on L°^{G).~~

~~(h) There is a net (fa)aeA in such that * fa - /a||i 0 {g e G). Anieiiability of Certain Banach Algebras 15~~

~~(i) There is a net {fa}a€A in such that ！!/*/«-/a||i 0 (/G I/i(GO�||=i)~~

~~(j) If G acts affinely on a compact, convex subset K of a locally compact convex space~~

E，

~~(i.e.) g-{tx + (l- t)y) = t{g • re) + (1 — t){g • y) [g eG,x,yeK,te [0,1])~~

~~such that G X K ==> K, {g,x) g • x is separately continuous, then there is x £ K~~

~~such that g • X = X \fg E G.~~

~~(k) Every irreducible unitary representation of G is weakly contained in A2.~~

~~(1) The trivial representation of G on C is weakly contained in 入2.~~

~~(m) There exists p e N such that for any compact subset K in G and e > 0, there is~~

~~f e LP{G) with ll/llp = 1 such that~~

~~114 * / - /II < 6 i'-^eK)~~

~~(Pp condition)~~

~~(n) For any p G N and for any compact subset K in G and e > 0, there is f G D\G)~~

~~with II/lip = 1 such that~~

~~Il�*/-/||< e (X e K)~~

~~(o) Iffe L'ioy, then ||Ap(/)|| = ||0||] for some peN.~~

~~(p) Iff then ||Ap(/)|| = ||0|ii/or a//p G N.~~

~~(q) C;(6；) ^~~

~~Theorem 2.1.6. Let G be an amenable group. If H be a closed subgroup of G, then H is also amenable. If N is a closed normal subgroup of G, then G/N is amenable.~~

~~Theorem 2.1.7 (Johnson's amenability theorem). Let G be a locally compact group.~~

~~Then G is amenable if and only if L^{G) is amenable. Anieiiability of Certain Banach Algebras 16~~

~~Proof. Refer to [Joh2] [Theorem 2.5). •~~

~~Definition 2.1.8. A Banach algebra A is said to be weakly amenable if A*) 二 {0}~~

~~Theorem 2.1.9. Let G be a locally compact group. Then is weakly amenable.~~

~~Proof. Refer to [D-G] [Theorem 1] or [Joh3]. •~~

~~Definition 2.1.10. A Banach algebra A is biprojective if A : A0A ——> A has a bounded~~

~~right inverse which is an ^-bimodule homornorphisin.~~

~~The relationship between biprojectivity and amenability is given by the following~~

~~lemma.~~

~~Lemma 2.1.11. Let A be a Banach algebra with bounded apprvximate identities. If A is~~

~~biprojective, then A is am,enable.~~

~~Proof. Refer to [Dal] [Proposition 3.3.20]. •~~

~~Theorem 2.1.12. Let G be a locally compact group. Then G is compact if and only if~~

~~L^(G) is biprojective.~~

~~Proof. Refer to [Dal] [Proposition 3.3.32] or [Run] [Example 4.3.2], [Lemma. 4.3.10] and~~

~~[Exercise 4.3.11). •~~

~~2.2 Amenability and weak amenability of M{G)~~

It is well known that two groups are topological isomorphic if and only if their measure algebras are isometric isomorphic. As a result, there is a 1-1 correspondence between the categories of locally compact groups and measure algebras. It is thus reasonable that the amenability properties of a locally compact group and those of its own measure algebra are closely related. It is proven that the measure algebra of a locally compact group is amenable as a Banach algebra if and only if the group is discrete and amenable. It is also true that the measure algebra is weakly amenable if and only if the group is discrete. In this section, we shall sketch the proofs of these two theorems. They are based on [D-G-H . Anieiiability of Certain Banach Algebras 17

~~Lemma 2.2.1. Let G be a non-discrete, locally compact group. Then Mc(G) has infinite~~

~~codimension in Mc(G).~~

~~Proof. The proof can be found in [D-G-H] [Theorem 2.7]. •~~

~~Theorem 2.2.2. Let G be a locally compact group. Then G is discrete and amenable if~~

~~and only if M{G) is amenable.~~

~~Proof. Assume that G is discrete, we have M{G) = L^{G) as Banach algebras. Therefore~~

~~M(G) is amenable by theorem 2.1.7. Conversely, if M{G) is amenable, then Mc[G) is~~

~~amenable since Mc{G) is a complemented ideal in M(G) = Mc{G) ® Ms{G). Therefore,~~

~~there exists a bounded approximate identity for Mc(G). By Cohen's factorization theorem,~~

~~we get MC(GY 二 M丄G). If G is not discrete, then Mc(G) has infinite codimension~~

~~in MciG) by lemma 2.2.1, which contradicts (*). Therefore G is discrete, and hence~~

~~M{G) = L^{G) is amenable, so G is amenable. •~~

~~Definition 2.2.3. A Banach algebra A is said to have weak factorization if span(A^) — A~~

~~Proposition 2.2.4. A weakly amenable Banach algebra A always has weak factorization.~~

~~Proof. See [B-C-D] •~~

~~Let Abe & Banach algebra. Let be a character on A. It is easy to see that C is an~~

~~^-bimodule under the products~~

~~a • z = z • a = (f){a)z {a e A, z e C)~~

~~This one-dimensional bimodule is denoted by~~

~~Definition 2.2.5. A derivation from A into C^ is called a point derivation at 4> if it, is a linear functional d on A such that~~

~~d(ab) = (l){a)d(b) + (l){b)d(a) (a,b e A)~~

~~Proposition 2.2.6. There is no non-zero, continuous point derivations on a weakly amenable Banach algebra. Anieiiability of Certain Banach Algebras 18~~

~~Proof. Let d : A——> C ^ be a, point derivation of ^ at 0 and define a derivation d^ :~~

~~A —A* by = d(^a)(f). Since A is weakly amenable, there exists ip e A* such that~~

~~d�a)(j) = ijj • a — a • Ip. For any x 6 A, we have~~

~~d{a)(f){x) = ip{ax) — ip(xa)~~

~~Thus, for any a E A, we have~~

~~d(a)^(a) = - ？々(0.2) = 0~~

~~By the definition of d, d{o?) = 2(f){a)d{a) = 0. Since A has weak factorization, it follows that d = 0. •~~

~~Definition 2.2.7. Let A 6 M{G)*, seG. Define s • A and A • s by~~

~~{/J,, s • A) = {ji^ Ss, X) and (/i, X- s) = {6s* (Jt； A) (/z G M(G))~~

~~Clearly,5 • A and A • s e M{G) and ||s • A|| = ||A • = |!A||. The functional A is said to be translation invariant ii s • \ • t = X {s,t E G)~~

~~Definition 2.2.8. By the decomposition M{G) = Mc{G) 0 Ms{Cr), let, 0 : M{G) > C,~~

~~J2seG H Then 小 is called the discrete augmentation character on M{G)~~

~~Lemma 2.2.9. Let G be a non-discrete, locally compact group. Then there exists a non- zero, translation invariani linear functional F in M(G)* such that~~

~~(a) F\M^{G) = 0~~

~~(b) FU^g)^ = 0~~

~~Proof. The proof can be found in [D-G-H] [Theorem 3.1]. •~~

~~Theorem 2.2.10. Let G be a locally compact group. Then the followings are equivalent:~~

~~(a) G is discrete.~~

~~(b) M{G) is weakly amenable. Anieiiability of Certain Banach Algebras 19~~

~~(c) M(G) has no non-zero, continuous point derivation at 4>-~~

~~Proof, "(a)力(b)" If G is discrete, then M{G) ^ L\G). Therefore, M(G) is weakly amenable by theorem 2.1.9.~~

~~"(b) (c)" See proposition 2.2.6.~~

“(c) (a)" Let F be as in lemma 2.2.9. It is sufficient to show that F is a point derivation at (f). Note that F|Md(G) 二 ^|MC(G)2 = 0 and F is translation invariant. Take ji G Mc{G), u 二 ZISSGOS^^S e Md{G), then

~~FUi * ") 二 E = a,F{s = …厂(,�= seG S€G seG~~

~~Similarly, = F{/i)(/){iy).~~

~~Therefore, for any G MC(G'), i/i,z/2 G Md{G),~~

F({/Ii + Ai2) * ("1 + "2)) = F{l.Li * //2) + F(JM * 1^2)十 F�丨丄2 * + F{ui *

~~=0(i/l)F(/V,2) + = + "l)F(/i2 + "2) + F{fli + + "2) Therefore, F is a point derivation at 0. • Chapter 3~~

~~Amenability and Fourier algebras with Bounded Approximate identities~~

~~Throughout, this chapter, let G be a locally compact group. We know that the Fourier~~

~~aigebra of G can be regarded as the "dual" object of its group algebra. However, in certain~~

~~cases, the Fourier algebra does not really behave like the group algebra. The major~~

~~difference between them is that the group algebra always has a bounded approximate~~

identity while the Fourier algebra does not. The proof of many important results for the group algebra depends on the existence of bounded approximate identity. As we will see in this section, Leptin proved that A{G) has a bounded approximate identity if and only if G is amenable. Thus, many important results for the group algebra can be transplanted easily to the Fourier algebra in the amenable case. Surprisingly, these results are in fact equivalent to the amenability of G.

~~20 Anieiiability of Certain Banach Algebras 21~~

~~3.1 Properties of Fourier-Stieltjes algebras, Fourier~~

~~algebras, group von Neumann algebras~~

~~The following results are quoted for future convenience. Their proof are omitted and may be founded in [Eym] unless specified.~~

~~Definition 3.1.1. Define P(G) := {/ G Cb{G) : f�Ag* ”) > 0 V.g G L^iG)}. The elements in P(G) are said to be positive definite function on G.~~

Definition 3.1.2. A unitary representation of C is a homomorphism tt from G into the group U{HT^) of unitary operators on some non-zero Hilbert space H-^ that is continuous with respect to the strong operator topology.

~~Definition 3.1.3. Let G be a locally compact group.~~

~~(a) Let Tjg be the set of all equivalence classes of unitary representations of G and G~~

~~be the subclass of all irreducible representations of G.~~

~~(b) Let A2 ： G > £(1.2(0), \2{x){g) (� *g [g e L\G)) be the left regular~~

~~representation of G.~~

~~(c) For any / e L\G), define~~

~~It is easily seen that || . ||c*(g) is a norm on L] (G). Let C*{G) be the completion of~~

~~LHG) under || •~~

~~(d) Let VN{G) be the von Neumann algebra generated by X2{G) in It is~~

~~called the group von Neumann algebra of G.~~

~~(e) For any f e define ll/llc； := ||A2(/)||~~

~~It is easily seen that || . is a norm on Let C*{G) be the completion~~

~~of LHO) under || • ||C;(G)- Anlenability of Certain Banach Algebras 22~~

~~Remark. By theorem 1.1.17, ),2 is really a unitary representation of G.~~

~~Frorn now on, write~~

~~• B(G):= {x f--t< 1f(x)~,'TJ >: 1f E ~G,~,'TJ E 7t7r }~~

~~For any function f on G, write~~

~~'l~heorem 3.1.4. Let G be a locally compact group. For any f E B(G), define~~

~~IIfIIB(C) := inf{II~III1'TJ1I : f(x) =< 7r(x)~,'TJ > where 1f E ~C,~ , 'TJ E 1{7r}~~

~~Then the following statements are tn.le:~~

~~(a) 11· lIe-(c) and Ii . IIc; 2(0) are C*-norm~ and 11 . IIc ~ (c) ::; 11 . IIct2(G) ~ I1 . IILI(G) on Ll(G)~~

~~(b) B(G) = span(P(G))~~

~~(c) (B(G),11 . IIB(G)) is a *-Banach algebra 'With pointwise multiplication and cornplex conjugation.~~

~~(cl) IIfIlB(c) = sup{1 Jc jgl : 9 E Ll(G), IIgllc-(c) ::; I}~~

~~(e) (B(G) , 1I·IIB(c)) ~ (Ll(G), 11 ·lIc-cc)) * via~~

~~< j,g >(£l(G),Bl(C)) = < 1f£l(c)(f)~,'TJ >= ler f(x)g(x)dmc(.x)~~

(f) (B(G) , II· IIB(C) ~ (C*(G), " . IIc*(c))*

~~(f E C*(G):g E B(G),g(x) = < 1fC(X)~ , 'TJ >,~,'TJ E 1t7r ) Anlenability of Certain Banach Algebras 23~~

~~(g) B( G) 1:S a translation invariant (generally non-closedJ-i:-subalgebra of LOO (G) and~~

~~11 . 11 B(G) ~ 11 . 11 L OO (G) .~~

~~Theorem 3.1.5. Let U E B(G)) x E G. Then~~

~~Theorem 3.1.6. Let G be a locally compact group. For any f E A(G)) define~~

~~IIfIlA(G) := inf {1I~11I17]1I : f(x) =< A2(x)~,r7 > where ~,7] E L2(G)}~~

~~Then the following statements hold:~~

~~(a) II·IIB(G) = II·IIA(G) on A(G).~~

~~(b) A(G) = L2(G) * L2(G)------~~

~~(c) (A(G), II·IIA(G)) is a closed ideal in (B(G), II·IIB(G)) .~~

~~(d) A(G)* ~ V N(G) via~~

~~< ~ * ij, T >(A(G),vN(G)):=< T~) 7] > (~ , 7] E L2(G), T E V'N(G))~~

~~and~~

~~(e) A( G) is a translation invariant *-subalgebra(generally non-closed) of Loo( G) and~~

~~11 . IIA(G) ~ 11 . liLOO (G).~~

~~(f) A(G) is the 11 . II B(G)-closure of B(G) n Cc(G) in B(G).~~

~~(g) CJ(A( G)) = G and hence A( G) is sem,i-simple.~~

~~Definition 3.1.7. (a) B(G) is called the Fourier Sti.eltjes algebra of G.~~

~~(b) A(G) is called the Fourier algebra of G. Anieiiability of Certain Banach Algebras 24~~

~~(c) C*{G) is called the full group C*algebra or simply group C*algebra of G.~~

~~(d) C*(G) is called the reduced group C*algebra of G.~~

~~Theorem 3.1.8. /Wiener- Tauberian theorem] Let G be a locally compact group. Then~~

~~for any g £ G, there exists u G A{G) such that u(g) / 0.~~

~~八 As G is a locally compact abelian group, it is known that G = {7 : G > T : 7 is a~~

~~continuous homomorphism}. Under the topology of compact convergence on G and the~~

~~pointwise product, G becomes a locally compact abelian group. In this case, we call G~~

~~the dual group of G.~~

~~Theorem 3.1.9. Let G be a locally compact abelian group. Then we have the following~~

~~statements:~~

~~(a) a{L\G))^G~~

~~(b) L\G) = A(G)~~

~~(c) M{G)兰 B{G)~~

~~(d) G^G~~

~~Therefore, for any locally compact group G, A{G) and B(G) are considered as the~~

~~“non-abelian" version of and M{G) respectively. Now, we have the following beautiful analogues of corollary 1.1.24 and theorem 1.1.25:~~

~~Theorem 3.1.10. Let G be a locally compact group. Then we have the following statements:~~

~~(a) Gi and G2 are topologically isomorphic if and only if A{Gi) and A{G2) are isomet-~~

~~rically isomorphic.~~

~~(b) G'l and G2 are topologically isomorphic if and only if B{Gi) and B{G2) are isomet-~~

~~rically isomorphic.~~

~~Proof. Refer to [Wal] [Corollary of Theorem 3]. • Amenability of Certain Banach Algebras 25~~

~~3.2 Conditions on A( G) that characterizing the amenabil ity of G~~

~~Proposition 3.2.1. let G be a locally compact group. Then~~

~~(a) Co(G) is a Banach A(G)-bimodule via the pointwise multiplication.~~

~~(b) M(G) is a Banach A(G)-bimodule via the module actions~~

~~d(p, . g) := d(g . p,) := gdp,~~

~~(c) Ll(G) is a Banach A(G)-submodule of M(G) via the module actions stated in (b).~~

~~Proof.~~

~~(a) : For any f E Co(G), 9 E A(G),~~

~~Ilfgl/oo = Ilgflloo ~ Ilfll oo llglloo ~ IlflloollgII A(G)~~

~~(b) : For any J-l E J\1(G) , 9 E A(G), f E Go(G)II'lloo~l'~~

~~lfa fd(g· J.L)I = lfa fgdJ.L1 ~ Ilf9II oo llJ.LIIM(G) ~ IIglloollJ.LIIM(G) :S IlgIIA(G)IIJ.LIIM(G)~~

~~(c): It trivially follows from (b) and the fact that L1 (G) is a closed ideal in M (G). D~~

~~V N(G) is a dual Banach A(G)-bimodule in a natural rnanner. Denote the norm closure of A(G) . V N(G) in V N(G) by UGB(G).~~

~~Remark. In the view of theorem 1.1.19, when G is abelian, UCB(G) is just the space of uniformly continuous functions on G.~~

~~Theorem 3.2.2. Let G be a locally compact group. Let A( G) be Fourier algebra. Then the following statements are equivalent.~~

~~(a) G is amenable.~~

~~(b) A( G) has a bounded approximate identity in the boundary of its unit circle. Anieiiability of Certain Banach Algebras 26~~

~~(c) A{G) has a hounded approximate identity.~~

~~(ci) Co(C)=鄰)• Co(G)~~

~~(e) M{G) = mAiG){Co(G),VN{G)) ^ 现•(網’ M(G))~~

~~(f) Li(G) 二 9Jt_4(G)04(GO，Li(GO)~~

~~(g) 5(G) .1(G))~~

~~(h) AiG) IS closed in -4(G')).~~

~~(i) The norm of A{G) is equivalent to that induced by the regular representation of~~

~~A{G).~~

~~(j) A{G) IS a{B{G),C*(G)) - dense in B{G).~~

~~(k) If A is a translation-invariant, conjugate invariant subalgebra of B(C) which sepa-~~

~~rates points of G, then A is a{B(G),C*(G)) — dense in B{G).~~

~~(1) The map N——> B{G/N) is a bijection between the set of all closed normal subgroups~~

~~N of G and the set of all weak*-dosed, invaiiant *-subalgebras A of B{G) with~~

~~A^O.~~

~~(m) UCB(G) = A(G) • VN(G)~~

~~(n) A{G) = A{G). A(G)~~

~~(o) A(G) = span(A(G). A(G))~~

~~Step 1: We shall show that "(a)力(b) (c) (d) (e) (f)々（a)，，~~

~~We first have the following lemmas.~~

~~Lemma 3.2.3. Let G be an amenable, locally compact group. Then, for each compact subset K oj G and e > 0； there is a f e such that * / - /||i < e (g e K). Anieiiability of Certain Banach Algebras 27~~

~~Proof. Fix fo e 丄 1(60�||=1 Sinc. e Ai : G —> Xi(g){f) ：= g ^ f (f e L\G)) is~~

~~a strongly continuous representation, there is a neighborhood U of ec in G such that~~

~~||^,*/o-/o||i = ||Ai(^)/o-/o||i <1 (1)~~

~~Since G is amenable, there is a net {fa}aeA in such that ||/ * fa — faWi — 0 (/eLi(G)『丨丨=1)~~

~~and thus, in particular,~~

~~P.*/o*/a-/.||i->0 {g e G) (2)~~

~~Since K is compact, there are G K such that K C Ugj.~~

~~By (2)，there is /«� G such that~~

~~\Sg*fo*fao -/olii < J (.7. = 1，2，…，n) and ||/o */«o "/o||i < j~~

~~so that, */o* fao - fo * /aolll < \ U = ]，2, u)~~

~~Let / = /o * fao, h G K. Then there is j € 1，2, ...，n such that hgj G U.~~

~~We then obtain:~~

丨丨W-/"I < IKg广 * ‘ */o* /ao -/o* /aolll < ；广 * ^gj */0* fao - ^hgj' */0* /aolll + * /o * fao — /o * /aolll =Pg广 */o* fao - kgj */o* /aolll + ll^/^^-i * fo — /o||l + ^ = e (geK) by (1) and (2) •

Lemma 3.2.4. Let {up)^ be a net in the unit ball of B{G) such that up —> uq in o{B{G), C*{C))-topology. Let (Eu)um be the hounded approximate identity o/Z/(G) stated in theorem 1.1.18 and let ey = Eu * EJ for any U eM. Then for any e > 0, there exists UQ and Po such that such that

~~Ikt/o *U0- uaWs^G) < e for any P > Po Amenability of Certain Banach Algebras 28 and~~

~~Il euo * Uo - 71,OIlB(G) < E~~

~~Proof. Assume Uo =1= O. Let C*(G) Ube the C*-algebra formed by adjoining 1 := 6ec to~~

~~C*(G). If f E Ll(G), then since ea* = ea, for any 71, E B(G), we have~~

~~1 < ea * u - u , f >(B(G),C"' (G)) 12 = 1 < 'U, (ea - 1)* * f >(B(G),C " (G)~) 12~~

~~::; IluIIB(G)1 < lul l (ea - 1) * f * f * * (ea - 1) >(B(G) , C * (G)~) 1~~

~~::; lIull B(G) IIf * f * 11 C" (G) 1 < 171,1, (1 - ea)* * (1 - ea) >(B(G),c"(G)a) I~~

~~Since 11 ea 11 c .. (G) ::; 11 ea III ::; 1, 0 ::; ea ::; ] in C'" (G)U. It follows that~~

~~Hence, we have~~

~~Note that lul(ea ) --+ lul(l) = lIuIlB(G). Let ao be such that~~

~~Then~~

~~Choose f30 such that lIupo11 « lupoI, 1 - eao >(B(G),C " (G)~)) < E if fJ ~ flo. By (3 .1),~~

Lemma 3.2.5. Let A c LOO(G) be a norm bounded and f E Ll(G). If (cPa)Q 'is a net in A such that cPa --+ cP in O"(L OO (G), Ll(G))_topology, then f * cPa --+ f * cP uniformly on compact subsets of G. Anieiiability of Certain Banach Algebras 29

~~Proof. Fix a compact subset K C G. For any x e G f G G ^> x L^f~~

~~is continuous, therefore [Lxf ： x G K) is a compact subset of Note that on norm~~

~~bounded subsets, the w*-topology is stronger than the topology of uniform convergence~~

~~on compacta. Hence,~~

~~f * Mx) =< 0a, At/�— < 0，LJ‘ >=/* Hx)~~

~~uniformly for x G K. •~~

~~Lemma 3.2.6. Let U G B(G) and let {ua)a be a hounded net in B{G) such that u^ — u~~

~~in a[B(G), C*{G))-topology. Then, for any gje C人G),~~

~~ll((/ * g) * - ((/ * g) * u)v\\AiG) — 0 (i;G A{G))~~

~~Proof. Let 7 = Let v G A{G)门 CJfi) and let K he a. compact subset of~~

~~G such that {suppf)~^{suppv) C K. Then for any w e L°°{G) and h e (G), we have~~

~~< (/ * �)t)’ ">(L°o(G),Li(G)) = < (vh) >(L-(G),L1(G))~~

~~=< ^XKJ* (”,0 >(L~(G)’L1(G)) = < (/ * {wXK))v,h >(L-(G),L1(G))~~

~~Note that /，(wxk)� eL^(G) which implies that f * G A(G) and~~

~~IK/ * w)v\\AiG)\\f * < WfhWwXKhMAiG)~~

~~Let w 二 ff * (Wa - u). By lemma 3.2.5, we have ^ * (wa - w) 0 uniformly on K.~~

~~Consequently, we get~~

~~\\[f*9Hu.-u)]v\\AiG) < ||/||2||b*K-«)XAni2||i'!U(G) — • (f,g G e A(G)n(7,(G))~~

~~Since < < ||/*<7||l'(G)7, ((/*(/)*Wa)a is a bounded net in B(G). It follows by the density of A(G) D C人G) in A(G) that~~

~~11 [/ * ” [u^ - 一 0 (/’ g e a{G), v e A{G)) • Anieiiability of Certain Banach Algebras 30~~

~~Lemma 3.2.7. Let S = {u e B{G) : = 1}. Let up e S and u e S. Then the~~

~~following statements are equivalent:~~

~~(a) U0 u in a(B{G), C*{G)) — topology~~

~~(b) U0 — u uniformly on compact subsets of G~~

~~(c) U0V uv in II . |U(G) {v e ^(C))~~

~~Proof.~~

~~"(a) (c)" Let e > 0 and U�b ea relatively compact neighborhood of e^, Va be a open~~

~~neighborhood of ec such that VQ, = and Let {eu)u& be the bounded approximate~~

~~identity of L^(G) stated in theorem 1.1.18. By lemma 3.2.4, there exists UQ, PQ such that~~

~~\\euo U0\\b(g) < f for any 0 > po~~

~~and ||eLo, * u - < e~~

~~Take F = e��i lemman . 3.2.6. Then, there exists j3i > PQ such that for any 0 > pi,~~

~~\\[euo * {up - U)]V\\b{G) < e~~

~~Thus, if V e A{G) and P > A, we have~~

~~II…卢—< 11^9—ec/o*"Hb(G)+||[et/o*(W/3-7z)]i;||B(G)+||(e[/o)*w—^tWls(G) < (2||v|U(G)+l)e~~

~~"(c) (b)” Suppose U0V uv in || •丨丨够）(v e A(G)), let K be a compact subset of G.~~

~~There exists v G A(G) such that v\k 三 1.~~

~~Then \\{Ua — w)X/<||oo < UK — — 0 Anieiiability of Certain Banach Algebras 31~~

~~“(b) (a)" By assumption, for any compact subset K C G, — U)XK\\OC — 0.~~

~~Let f E CC(G). Then~~

~~I < Ua-Uj >(^BiG),C*{G)) | < / = / \ua-u\\f\ < ||/||l || C^a—卿p(/) lloo�.• Jg Jsuppif) Note that supa\\ua — ？< 2 and Cc{G) is dense in C*{G),~~

~~Lemma 3.2.8. Let G be a locally compact group. Then~~

~~mAiG)iCo{G),VN{G))兰 mAiG){A[G),M[G))~~

~~Proof. For any (I> G yiV(G)), define F^ G £.{A(G),M(G)) by~~

~~< /, r^{v) >iCo{0),M{G)))--^(^VN{G),A{G)) (/ ^ Co(G), V e .4(C7))~~

~~Since for any f e Co(G),v,g € A{G)),~~

~~< /’ 'v)>=< -v>== < ^(g . /), >二 < g . f, >~~

~~We have r中 G dJflAiG){A{G), M{G)). Thus,~~

~~二 sup {||r^?;)||M(G) ： V 6 乂(GOll.iisi}~~

~~二 SUP {I < /，�(Co(G)’M(G)) )I : e A(G')||.||<1 ’ f G Co(G')||.j|~~

~~==sup {| < >(VN(G)，M(G)) I ： V G A(G)ii.ii~~

~~=sup {mmvNiG) •• f e ==li 到 |£(Cb(G)’WV(G))~~

~~Therefore, F^, is an isometry.~~

~~Conversely, for any T e ^a{G){A{G),M{G)), define e 2.[Cq(G)N{G)) by~~

~~< V >(vyv(G),.4(G)):= < /，r>) >Co(G),M(G)) (/ e Cq(G), V e A(G)) Amenability of Certain Banach Algebras 32~~

~~Similarly, ~r E 9RA(G )(Co(G) , V N(G)).~~

~~It is eaBily seen that~~

~~and r f-t is an isometric isomorphism. o~~

~~Remark. For any 1 E Ll(G), U E A(G) , 9 E Co(G) ,~~

~~So, for any 1 E L1(G) , ).. 2(1) E £(A(G) ,L1(G)) and~~

~~II A2(1)11£(A (G),Ll (G») = sup{llu· A2(1)11 : u E A(G) , Ilull oo :::; I}~~

~~Lemma 3.2.9. Let G be a locally compact group. Let X := {h E Co(G) : h(x ) =~~

~~2:: 1U i(X)gi(X),Ui E A(G),gi E C~(G) , 2: : 1I1u i Il A ( G )llgll co ::; oo}. Define the norm~~

~~IIhll x := in f{2: ~:1 1I ui II A(G )llgi ll oo : h(x ) = 2:: 1U i(X)gi(X):,Ui E A(G),gi E Co(G)}~~

~~(h E X). Th en 9RA(G) (CO(G) , V N(G)) ~ X * and the duality is .Qiven by~~

~~co <~~

~~h > == L < (VN(G),A(G») i=1 co (h E X, h(x ) = L Ui(X)g i(X), 'Ui E A( G) , gi E Co( G) ,~~

~~Proof. Write 9R := 9RA(G) (Co(G) , V N(G))) for convenience. Let F E X *. For any U E A(G) , 1 E Co(G), define~~

~~< (V N(G),A(G»: = F(ul)~~

~~Since~~

~~, we have 11~~

~~Also, for any 1L, v E A(G), f E Co(G) ,~~

~~(VN(G),A(G) = F(u(vf)) = F((uv)f) Amenability of Certain Banach Algebras 33~~

~~=< iPF(f), uv >(VN(G) ,A (G»=< V· (VN(G) ,A(G»~~

~~which implies that~~

~~Conversely, let~~

~~00 Fcp(h) := L < iP(gi ), Ui >(VN(G),A(G» i=l~~

We have to show that (h) does not depend on the representation of h.

~~Let {ea}aEA be an approximate identity of Co(G) s11ch that {ea}aEA ~ A(G) and~~

~~S'UPaEAllea ll oo ::; 1. Let E > °and choose N such that~~

~~00 L Ilui II A(G)llgill oo < E i=l~~

~~and choose ao E A such that~~

~~00 L IluiIIA(G) IIgieao - gi ll oo < E i=l~~

~~Then~~

~~IFcp (h)1~~

= I 2:~1 < (VN(G),A(G» I ::; 12::1 < (gieao - gi ), Ui >(VN(G),A(G» I + I I: ~1 < (gieao ), Ui >(VN(G) ,A (G» I

::; 2:~1 1I(gieao - gi ) II VN(G) IIUi Il A(G) + I I::1 < eao . iP(gi )' Hi >(VN(G),A(G» I ::; 1IlIryn 2: ~ 1 Ilgieao - gi)lIoolluiIlA(G) + I I::1 < (gi)' eaoUi >(VN(G),A(G» I ::; 11 11 ryn(E + 2E) + I I::1 < gi, rcp( eaoUi) >(Co(G),M(G) I

~~::; 3EIIlIryn + I I: ~l < gi, Ui . r cp (eao ) >(Co(G),M(G» I~~

~~::; 3EIIlIryn + I I: ~ l < Uigi, rcp(eao ) >(Co(G),M(G» I~~

~~::; 3EIlcI>lIryn + I < h , rcp( eao ) >(Co (G),M(G» I~~

~~Therefore, if h = 0; then F(h) = o. Hence, the claim is proven.~~

~~Let IIhll x ::; 1, E > O. Choose gi, E Co(G), Ui E A(G) such that h = 2::1Ilu III A(G) lIgi Il 00 ::;~~

~~IIhllx +E. Then IF(h)1 ~ 1IlIryn I:~l IIgi ) 11 00 IIUi IIA(G) = 1I~~

~~We may now continue "Step 1".~~

~~Proof.~~

"(a) (b)" Let {fa}a&A be a net in such that ||� /*« - /a||i 0 uniformly 1 ^ on compact subsets of G. Let ^^ (/a)泛’ and e�：二 *

~~We claim that: {e^jaeA is a net in 乂(G)||.||=] C B{G)\\.\\=i~~

~~and Ca —» 1 uniformly on compact subsets of G.~~

~~"Proof of the claim:" Note that~~

~~= ll^a * 4|U(G) < (|fe||2)2 = (ll/alll) 二 1 and~~

~~||ea|U(G) = sup{\ < re^，《e�>:T eVW(G)} > I < >|-|<^a,Ca>| -二 ||/al|l 二 1~~

~~The first statement follows. For the second statement,~~

~~=\ JG M工y�”cXy)‘'dy - i|~~

~~=\ Jg fc^(列- fc fa[xy)dy\~~

~~1 < 1 • Wfa - ^x* faWi —^ 0 uniformly on compact subsets of G.~~

~~This, however, is already sufficient for {ea}a to be a bounded approximate identity for~~

~~A{G) by lemma 3.2.7. "(b) (c)" Trivial.~~

~~"(c)玲(cl)" By Cohen's factorization theorem, the result follows. Anieiiability of Certain Banach Algebras 35~~

~~"(d) ^ (e)" It follows from lemma 3.2.9.~~

~~"(e)冷(f)" Note that Li(G')) C mAiG)(A(G), M(G)) = M{G). For any r G there exists e M{G) such that r(u)=邓 and ufi 6 L\G)~~

~~{u e A{G)). Let E be a Borel set in G such that mc^E) = 0. Then~~

~~I d{u . ij,){x) = f u(x)dfj,{x) = 0 (ue A(G)) JE JE ‘~~

~~For any compact subset K C G, there exists f € A{G) such that 0 < / < 1 and / = 1 on K. Thus~~

~~"(K nE�= f XK{x)dii(x) < [ u(x)dfi{x) = 0 Je Je By regularity of /i, we have ii[E) = 0. As a result, ji is absolutely continuous with respect to rriG, and so G~~

~~” (f) (a)” Uge f e L\G\ u e A{G\~~

~~||A2(^/)^?||2 = \\uf * < Ilk/1 * Mlh < IIIMIocl/l * Mlh < |MUoi|A2(|/|)|i||.9l|2~~

~~Therefore,~~

~~ll'u-A2(/)lk_(鄉= ||A2W)|| < IMU|A2(/)II life L\Gy, then l|A2(/)lk_M(G)，LnG))引|A2(/)|| (3.2)~~

~~If (f) holds, the norm || • ||i of L乂G) is equivalent to || • Combining with~~

~~(3.2)，there exists a positive number K such that~~

~~ll/lli < I|A2(/)|| {feL\Gr)~~

~~By replacing / by f * f\ we have~~

~~if m? < M\\x2U)r JG Anieiiability of Certain Banach Algebras 36~~

~~By induction,~~

~~(/fix)) < VnGN JG Therefore, we have | < ||入2(/)!|. By theorem 2.1.5, G is amenable. •~~

~~Remark. Lemma 3.2.3 is well-known and can be founded in [Run] [Lemma 7.11] for~~

~~example. The remaining theorems and lemmas in the above are from [Fig], [Gra-L], [Neb]~~

~~and [Lau].~~

~~Step 2: We shall show that "(a) (g) (h) (i)~~

~~Proof.~~

~~"(a) (g)” Let T E m{A[G), A(G)), and let {ea}«eyi be a,n approximate identity with~~

~~^ M with M > 0. Note that {Tea}aeA may be regarded as linear functionals~~

~~on €"(0), by bouridedness of T, has a C*(G))-limit point w G B(G).~~

~~Let {Teisjp^B be a subnet of {Te^ja^A such that Te/j -> w in a{B{G), C* (G)) - topology.~~

~~Fix u e .4(6'), note that~~

~~I [ {UTE曰-_)(/)l < IMW I {TEFS —义…(/)| - 0 (/G L\G)) JG JG~~

~~Therefore, by density of L、G) in C*(G) and the boundedness of the net {wre卢一it w}供仏 we get~~

~~uTep uw in a(B(G),C*{G)) - topology.~~

~~By boundedness of T again,~~

~~Tu = lim.aT{uea) = HmauT{ea) 二 uiu~~

Finally, we get T = L也 for some w e B{G) where L^ : A{G) > A{G),u ^ uw By theorem 3.1.8, if uwi = uw2 {u G A(G)), then Wi(g) == W2(g) (g G G). It, follows that there exists a unique w G B{G) such that T = Lyj. Clearly ||Tw|| < implies that ||T|| .< ||ii/||. On the other hand, TE^ — w in G(B{G),C*{G)) - topology and Anieiiability of Certain Banach Algebras 37

~~5(G)||.||<||r|| is (j{B(G),C*{G)) - closed, we have \\iu\\ < ||T||. Therefore, 8{0)>~~

~~dJt{A(G),A{G)),iu I— Lw is an isometry.~~

~~"(g) (h)" This is trivial since A{G) is always a closed ideal in B(G).~~

~~"(h) (i)" If (4(G)，II • II) is closed in M(A(G),A(G)) , then the norms ||. |U(G) and~~

~~II • IIfm(/1(G),/1(G)) are equivalent by open mapping theorem.~~

~~"(i) (a)" Refer to [Los2] [Theorem 1].~~

•

~~Remark. The proof of “ (a) (g)" is founded in [Ren .~~

~~Step 3: We shall show that ”（a) ^ (j) and (a) (1) (k)”~~

~~We quote the following lemma before continuing to proof the main theorem for con-~~

~~venience.~~

~~Lemma 3.2.10. Let G be a locally compact group. Let A he a weak*-closed, invariant subalgebra of B(G). Suppose A is conjugation invariant and separates the points of G.~~

~~Then A contains A{G).~~

~~Proof. Refer to [B-L-S] [Theorem 1.3]. •~~

~~We may now proceed “ Step 3".~~

~~Proof.~~

~~"(a) (j)" Let {ea}a€A be a bounded approximate identity in Then~~

~~ec^w -^w in \\- (W G A{G)) Ainenahility of Certain Banach Algebras 38~~

~~By lemma 3.2.7,~~

~~ecv 1 in a{B{G),C*{G)) - topology~~

~~For any v 6 B[G), note that Q and thus~~

~~I / (优� —…(/)l < IMU [ {ea - 1)(/)I ->0 (/ e L\G)) JG JG Since {vCa — v)a is a bounded net, it follows that~~

~~vea — V in the A(B{G),C*(G)) topology~~

~~” (j) zz> (a)" Since A(G) is a{B{G), C*{G))-dense in B{G), there is a net {e«} C A(G) such that ea — 1 in cr{B{G),C*{G)) — topology. By lemma 3.2.7，~~

~~eaW -> w in II . ||yi(G) (w e A(G))~~

"(a) =>• (1)" Suppose G is amenable, and let ^ 0 be a weak*-closed invariant *- subalgebra of B{G). Then N — {x e G \ ^ ^ = ^ V^ 6 A} is a closed normal subgroup of G. Moreover, A viewed as- a subalgebra of B{G/N) separates the points of.

~~G/N. By lemma 3.2.10 AiG/N) C 式.Since GjN is amenable by theorem?? and "(a)~~

~~(j)", A[GIN) is weak*-dense in B{GIN) and hence A = B{G/N). Therefore, the map is well-defined and surjective. If B{G/N) •— B{G/N'), then~~

~~= B{G/N)} = {x e G : ：^ ip = if y^p e BiG/N')} 二 N'~~

~~Therefore, the map is bijective.~~

~~"(1) (k)" Let A be the a(B(G), C*(G)) - closure of A(G) in B(C). Note that~~

~~A(G) is point separating and translation invariant, and so is A. If A B{G/N) for some non-trivial closed normal subgroup N < G^ then A cannot separates points in N. This forces A = B{G).~~

，，（k) (a)" Note that A[G) is a translation-invariant, conjugate invariant subalgebra of B{G) which separates points of G, so X D A{G) is a[B[G), C*{G)) - dense in B{G). • Amenability of Certain Beincich Algebras 39

~~Remark. All of the proofs in this step can be founded in [B-L-S .~~

~~I Step 4: We shall show that "(a) (m)”~~

~~We have the following lemma which is essential in this step. It is just [Lau] [Proposition~~

~~4.4].~~

~~.八 Lemma 3.2.11. Let G be a locally compact group, then UCB{G) is a subspace ofVN{G)~~

~~containing C*{G).~~

~~Proof. Suppose f € Cc(G),and supp{f) C K for some compact subset K C G. Let~~

~~(j) e A{G) such that (t){t) = 1 for all t e K. Then 0 •入2(/)=入2(/). Therefore,~~

~~X2{L\G)) C UCB{G) which gives C;{G) C UCB{G). •~~

~~We may now proceed “ Step 4".~~

~~Proof.~~

~~"(a) (m)" Since "(a) (b)", has a bounded approximate identity, by Co- hen's factorization theorem, the result follows.~~

~~"(m) (a)" Define~~

~~f ： AiG) X VN(G) —> UCB{G), (u’T) ^u-T~~

~~This induces a bounded linear mapping j : A{G) 0 VN{G) > UCB{G).~~

Let j* : UCB{Gy > (g) l/iV(G))* ^ S1{A(G), VN{GY) be its Banach adjoint. By assumption, j is onto. It follows tha,t there exists a real number c > 0 such that ||j*(/)|| > c||/|| for all f e UCB{Gy. It is not hard to see that r(UCB{GY) D m{A{G), A(G)).

~~Recall that A{Gy ^ VN(G) and UCB(G) C VN(G). Define i : A(G) 一 UCB{Gy be the restriction map. Since C*{G) is cr-weakly dense in VN{G), C*(G')||.||~~

~~UCB{d) 2 C:{G), which implies that i : A(G) 一 UCB{GY is an isoinetry. Now~~

~~j* oi: A(G) 一 dn(A(G),A(G)) C £(4(G)’ Anieiiability of Certain Banach Algebras 40~~

~~is just the canonical embedding. Therefore, {j* is closed in ^)Jl{A{G), A{G)).~~

~~Since “ (a) (f)", G is amenable.~~

•

~~Remark, "(a) (m)" is just [Gra] [Proposition 1] and "(m) (a)'"' is referred to [L-L~~

~~Proposition 7.1].~~

~~Step 5: We shall show that "(a) (n) (o) (a)”~~

~~There are several lemmas before finishing the proof of the main theorem which can be~~

~~founded in [F-G-L .~~

~~Definition 3.2.12. Let B he a Banach algebra of complex-valued continuous functions on a topological space. B is said to be weakly self-adjoint is there exists a KQ > 0 such that~~

~~\f\'eB and |||/|||B~~

~~Lemma 3.2.13. If B is a Banach algebra of complex-valued continuous functions on a topological space with weak factorization, then there exist N，K G N such that~~

~~寧’ K) ：= {F E B ： F = ^211 FI WITH Y^ n II//II ^ ^11/11} t=l j=l i=l j=l is dense in B.~~

~~Proof. Clearly, B = Un=i B(N,K). By the Baire category theorem, there exist~~

~~K' en such that B{N'.+ Thus there exists /o G B{N', K'), 6>0 such that~~

~~B{N\K')D{feB:\\f-fo\\<6}. Write fo = n;=i fi with UU 丨丨別 ^ ^1l/oll where [f悠^ 双 j = 1,2,3,4.~~

~~Let cj •— e^ and // = ujf'l then~~

~~2N' 4 -/o- E n 斤 Amenability of Certain Banach Algebras 41~~

~~and 2N' 4 2N' 4 L IT Ilf'1l1 = L IT IIf!1I ~ K'lIfoll i =l+N' j = l i =l+N' j=l Let f E B be such that 5/2 :s; IIfll :s; 5. Then lI(f + fo) - foil < 0 implies that~~

~~f + fo E B(N', K'). For any E E [0 , 5/4), there exist (f!)~l ~ B, j = 1,2,3,4 such that~~

~~lI(f + fo) - L: ~ 1 rr;=l fIll < E and~~

~~N' 4 LIT IIf! II :s; K'(lIfll + IIfoll + E) :s; K'(5 + IIfoll + E) ::; K'(25 + IIfoll) i=l j = l~~

~~Now, we obtain 2N' 4 N' 4 IIf - LITf! 1I = IIf - (-fo + LITf!)1I < E i=l j = l i=l j=l~~

2N' 4 LIT IIf!11 ~ K'(25 + IIfoll) + K'lIfoll = 2K'(lIfoll + 5) i=l j = l and 2N' 4 11 LITf!1I ~ 5/2 - E 2:: 8/4 i=l j = l We set N == 2N' and K = 8K'(lIfoll + 5)/5, then L::~' n;=l /If! 1I ~ KI/ I:;~; rI;=l fIll · Thus, f E B(N, [(). Since tB(N, K) = B(N, K) for all t > O. We see that B = B(lV, K). o

~~Lemma 3.2.14. Let B be a weakly self-adjoint Banach algebra of complex-valued contin·· uous functions with weak factorization. Let s(B) := {x : f(x) f 0 for some f E B}.~~

~~Then there exists K-1 > 0 such that for each compact subset M ~ s(B) and any h E B such that If I > 0 on M , there exists f2 E B such that~~

~~Proof. By lemma 3.2.13, there exist N , KEN such that B(N, K) = B. Thus, there is a f { E B(lV, I() such that Alnenabili ty of C rtain Banach Algebras 42~~

~~Write I{ = ~~ ] rr;=l f: with L:~1 rr;=l IIf: 11 ~ Kllf{ll· Without loss of generality, we may assllme that~~

~~Set I~ = 1/4 ~~1 rr;=l 11:12, so f - 2' ~ O. By the weak self-adjointness, we have f~ E B and~~

~~Ilf~1I ~ (2K)1/2 N KollfI 111/2~~

~~Note that~~

~~N 4 N 4 N 4 12 12 f~ ~ LIT 11111/2 ~ (L IT If:1)1/2 ~ I LIT f:1 / = If{1 / i=l j = 1 i= l j=l i=l j = l~~

~~Now 111(x ) - 1{(x)1 ~ IIf1 - f{11 ~ ~lli (x)1 for all x E M, so If21 ~ 1f{11/2 ~ (~)1/21.f111/2. Take 12 = ~.f~, K1 = 2K1/2N Ko. Then f2 ~ I.fl11/2 and~~

~~11 f211 ~ 4/3(2K) 1/2.N Ko 11 fI 111/2 ~ K111h 11 ]/2~~

~~Lemma 3.2.15. Let B be a weakly self-adjoint Banach algebra of complex-valued co.rd'ln-·~~

~~uous functions, and let s(B) := {x : f(x) f 0 for S01ne I E B} Suppose that B has~~

~~weak factorization. Then there exists K1 > 0 such that for each cornpact subset M ~ s(B)~~

~~there is an element I E B such that f ~ 1 on M, f ~ 0 on s(B), and Ilfll ~ .Kl·~~

~~Proof By ass'umptions on B and M, there exists f1 E B such that f] 2:: 1 on NI . By~~

lernma 3.2.14, ther are .f2,f3 .. . E B such that Ifj+l(x)1 ~ Ifj(x)11/2 for any x E M and Illj+111 ~ K11IfJI11 /2. Hence, Ij+ 1 > 1 on M and Ilfj+111 ~ KFi=o2 -il lhI12 ,-j . Choose j K2 = 2Kl and j E N such that Illj 112- ~ 2. Then f = fj+1 will be the required 1 EB. o

~~V-Ie may now proceed "Step 5" . Anieiiability of Certain Banach Algebras 43~~

~~Proof.~~

~~，，（a) (n)" Since "(a) (b)", A{G) has a bounded approximate identity, by Cohen's~~

~~factorization theorem, the result follows.~~

~~"(n) (o)" Trivial.~~

~~，，(o) (a)"~~

~~We first claim that: | /〇^/(0：)^/爪0(0：)| < ||入2(")|| {g e L\G)+)~~

~~"Proof of the claim:" By lemma 3.2.15, there is M > 0 such that for each compact~~

~~subset K C G there is an element f e A(G) such that f > 1 on K, f > 0 on G, and~~

~~11/11 糊 < M.~~

~~Clearly, for any g e LH^V. p2(州！ is equal to the norm of the linear functional h h f^h{x)g(x) on .4(G). Therefore,~~

~~I f f{x)gix)dmGix)\ < \\X2{9)\\\\f\\AiC) JG~~

~~[ ff�dmcy(x)引 f f{x)gix)dmG(x)\ < M||A2(^)|| JK JG By regularity of Haar measure,~~

~~f 9(x)dmG{x) < M\\X2{g)\\ JG~~

~~By replacing g by g * g*, we have~~

~~([9{x)f < M||A2�||2 JG~~

~~We hence obtain by induction that,~~

~~(F 9(X)) < MI/2IA2�I Ifor all rz e N JG~~

~~Therefore, we have | 办)l < 11入2(")||. By theorem 2.1.5, G is amenable. • Anieiiability of Certain Banach Algebras 44~~

~~Remarks. 1. The proof of "(o) (a)" is a blend of [Losl] [Proposition 2] and [Her]~~

~~Lemma 4 .~~

~~2. There are only a few groups having amenable Fourier algebras. Let G he a. locally~~

~~compact group and A{G) be its Fourier algebra. Then A{G) is amenable if and only~~

~~if G has an abelian group of finite index [F-R]. Therefore, amenability is a rather~~

~~restrictive requirement for Fourier algebras. Chapter 4~~

~~Operator Amenability and Fourier algebras~~

~~4.1 Operator Amenability~~

~~Before the discussion of the main theorems, we need some preliminary. Those results~~

~~presented below can be found in [Rual] unless specified.~~

~~Definition 4.1.1. A matrix norm, || • || on a vector space V is an assignment of a norm~~

~~II • oil the matrix space M„(l/) for each n G N. An operator space is a vector space~~

~~together with a matrix norm || • || for which~~

~~1. Ik ® ^llm+n = max(||叫|„，IHInJ~~

~~2. \\avP\\n<\\a\\\\vU\P\\ for all V e w e a e 礼’爪，(3 G M爪，N, m’n e N.~~

~~Definition 4.1.2. Given two operator spaces V and W and a linear mapping : V >~~

~~W, for each n 6 N, there is a corresponding linear mapping 小几M„(V) � defined by~~

~~0„和）:=[c}>{vi^j)] {v = [Vi,j\ G~~

~~45 Anieiiability of Certain Banach Algebras 46~~

~~We define the complete bounded norm, of • by~~

~~\\cf>U:=sup{UJ:neN}~~

~~We say that~~

~~• 0 is completely bounded if < oo~~

~~• 0 is completely contractive if ||0||c6 < 1~~

~~• is a completely isometry if each is an isometry.~~

~~• V and W are completely isoinetrically isomorphic if there is a completely isometri-~~

~~cally isomorphism from V onto W. In this case, we write V =c.i. W~~

~~Denote by CB{V, W) the vector space of all completely bounded linear maps from V to~~

~~Theorem 4.1.3. For each Hilberf, space H, every subspace W of is a operator space~~

~~where the mMrix norm is inhente.d from. =~~

~~Conversely, if V is a operator space, then there exists a Hilbert space H, a concrete operator space W C and a complete isometry ^ of V onto W.~~

Theorem 4.1.4 (Arveson-Wittstock extension theorem). IfV is a subspace of an operator space W, and H is a Hilbert space, then any complete contraction 小:V > £(7^) has a completely contractive extension $ ： W ~>

~~Definition 4.1.5. An operator space system V on a Hilbert space ‘H is a norm closed linear subspace V C £,{H) which is self-adjoint, i.e. x e V \i and only if x e V'% and unital.~~

~~Definition 4.1.6. Given operator systems V and W, a linear mapping 4> •' ^ ^ ^ is said to be completely positive if > 0 for all n G N.~~

~~Theorem 4.1.7. If (f) : V'——> W is a linear mapping of operator systems such that~~

~~(p{I) ~ I’ then (p is completely positive if and only if (j) is a complete contraction. Anieiiability of Certain Banach Algebras 47~~

~~Proposition 4.1.8. Let V, W be two operator spaces. For any v e Mn{V 0 W), we~~

~~define~~

~~IMIop ：= in/{|M||M||HI_| : u = a{v(^w)p,ve M,{V),w e M,{W),ae M„，购，G M•夠’„.}~~

~~Then {V~~

~~tensor product to be the completion of this space.~~

~~Definition 4.1.9. Let A be an operator space. A bilinear map m : A x A > A is said~~

~~to be completely contractive if it determines a completely contractive linear map~~

~~m : A§)A ~~> A~~

~~i.e. \\[aM\\ < ||a||||6|| {a = [a,,,-] G Mm{A) ,6= [bk,i] e Mn{A)，m’72 G N)~~

~~Proposition 4.1.10. Given two operator spaces V, W, then~~

~~Definition 4.1.11. Let be a Banach algebra which is also an operator space. A is said to be completely contractive if the multiplication of it is a completely contractive bilinear map.~~

~~Definition 4.1.12. Let A be a completely contractive Banach algebra. An operator~~

~~A — bimodule E is an A-bimodule which is also an operator space such that the module actions~~

~~Ax E >• E, (a, iT) a - X and E x A——> E, {x, a) t~> x • a are completely bounded.~~

~~Proposition 4.1.13. Let A be a completely contractive Banach algebra and let X be an operator A-bimodule. Denote by A^ the unitization of A. Then X is a operator A^- bimodule by setting~~

~~I • X = X and x • 1 = x {x G A)~~

~~二八 Moreover, A'S>X is completdy isometncally imbedded into y4”§X. Anieiiability of Certain Banach Algebras 48~~

~~Proof. Note that A^§>X ^ �i； an^ d the left module multiplication rrii : yl^^gX >~~

~~X corresponds to the complete contraction~~

~~(A§)X) 01 X ~~^ X, + ^ (a e A, x,p e X)~~

~~The case of the right module multiplication is similar.~~

~~Let TT : A^—— >A be the complete quotient map and i : A —-�A ^the natural injection. Then is decomposed into~~

~~Hence, for any u 6~~

~~IMI < ||7r0ic)x||||(«®ii)A')^ll < \\u\\~~

~~IK漸= IMI •~~

~~Definition 4.1.14. Let A be a completely contractive Banach algebra. If ^ is a operator~~

~~^-bimodiile, then is it not. hard to show that E* becomes a. operator ,4-bimodule through~~

~~< . a >:=< a . 00,(1) < x,a' (f) >:=< x • a,(f) > {a e A,x e E,(/) e E*)~~

~~The dual space of an operator bimodule equipped with these module operations is called~~

~~dual operator bimodule.~~

~~Definition 4.1.15. A completely contractive Banach algebra A is said to be operator-~~

~~amenable if for every dual operator /1-bimodiile E, every completely bounded derivation~~

~~D : A——> E is inner.~~

~~Proposition 4.1.16. Let A be a completely contractive Banach algebra, and let E, F be~~

~~operator A-bimodules. Then E®F becomes an operator A-bimodules through~~

~~a-(x^y) = (a- x) 0 y and {x ^ y) • a = x 0 (y • a) (a e A,x e E,y e F) Anieiiability of Certain Banach Algebras 49~~

~~Definition 4.1.17. If ^ is a completely contractive Banach algebra, then the diagonal~~

~~operator is a completely bounded ^-bimodule homomorphism which is defined by~~

~~Ayi : A^A ~~A,a<^b ab~~

~~Definition 4.1.18. Let A he EL completely contractive Banach algebra.~~

~~(a) An element M e {A(S)A)** is called a virtual operator diagonal for A if~~

~~a - M = M • a and a • A^M = a (a G A)~~

~~(b) A bounded net {ma}a£A ^ is called an approximate operator diagonal for A~~

~~a . iria - nia • a 0 and aA^ma 0 (a G A)~~

~~Theorem 4.1.19. An operator amenable completely contractive Banach algebra has a~~

~~bounded approximate identity. For a completely contractive Banach algebra A^ the fallowings are equivalent:~~

~~(a) A is operator amenable.~~

~~(b) There is a virtual operator diagonal for A.~~

~~(c) There is an approxiamte operator diagonal for A.~~

~~Proof. Refer to [RiialJ [Theorem 2.4]. •~~

~~Definition 4.1.20. Let E be an operator space. A closed siibspace F of E is said to be completely weakly complemented in E if there is a completely bounded projection from~~

~~E* onto Fi.~~

~~Theorem 4.1.21. Let A be an operator amenable, completely contractive Banach algebra and let I be a closed ideal of A. Then the followings are equivalent:~~

~~(a) I is operator amenable. Anieiiability of Certain Banach Algebras 50~~

~~(b) I has a bounded approximate identity.~~

~~(c) I IS completely weakly complemented.~~

~~Proof. Refer to [F-W] [Theorem 8.5] and [Wool] [Theorem 3]. •~~

~~4.2 Hopf-von Neumann algebras and their predual structures~~

~~Our sources of this section are [Rua2] and [Tak .~~

~~Definition 4.2.1. Given two von Neumann algebras M and acting on Hilbert spaces~~

~~K, respectively, the von Neumann algebra tensor product is defined as the von~~

~~Neumann algebraic acting on the Hilbert space tensor product H fC generated by the~~

~~algebra tensor product ！DT 0 9T.~~

~~Theorem 4.2.2. Let dJl, be von Neumann algebras with preduals WL and 0"t*，respec-~~

~~tively. Then we have a canonical completely isometric isomorphism~~

~~mM'yi. = (TOOT),~~

~~Proof. Refer to [Riia2] [Theorem 7.2.4]. •~~

~~Definition 4.2.3. A Hopf-von Neumann algebra is a pair (OJt, F*), where is a von~~

~~Neumann algebra, and F* is a co-multiplication, i.e. P is a unital, injective, normal,~~

*-homomorphism from dJfl to which is co-associative,. i.e.

~~r* r*®iDOT~~

~~on麵朋麵麵 Remarks. Anieiiability of Certain Banach Algebras 51~~

~~(a) Let r* : VJl——^ dJl'^W l be a unital, injective, normal, *-homomorphism. Since P~~

~~is w*-continiioiis, F* = (F)* for some T : 窃现)* ~> (肌)*. Note that P is a~~

*-homomorphism, F* is completely contractive, so is F :=~>

~~Thus the bilinear map F : x ~~>• is completely contractive. For~~

~~any f,g£ 971*，define~~

~~forg:=nf�g)~~

~~(b) < {r 0 iDan)(r» > 二< (r0/?.)，>~~

~~二< (/or")®"，r^>�~~

~~=< r((/�r~~

~~二< (/。“）Or K(f)>~~

~~Similarly, < f 0 g (S> "，{idm ^ r*)(P(/)) > = < /��（ g��h),(j) >~~

~~Therefore, op is associative if and only if T* is co-associative. Thus, (OJt*,or) is a.~~

~~completely contractive Banach algebra provide that (Wl, F*) is a Hopf-von Neumann~~

~~algebra.~~

~~4.3 Operator Cohomological properties of A(G)~~

Of course, someone may ask if there is any analogue for the Johnson's cohomology vanishing theorem for Fourier algebras. As proved in section 3.1, the amenability of the group does not imply the amenability of the Fourier algebra but only the existence of bounded approximate identity. In fact, A(G) is amenable if and only if G has an abelian subgroup of finite index [F-R]. Therefore, the original definition of amenability is considered as

~~"unsuitable" for the study of Fourier algebras. In [Rual], Riian, however, found a good notion of “amenability" for the Fourier algebra which is called "operator amenability".~~

~~He proved that G is amenable if and only if A{G) is operator amenable.~~

~~Let G be a locally compact group. From now on, use the following notations: Anieiiability of Certain Banach Algebras 52~~

~~• Define ！^,…。）：一 VN(G)'^VN{G) ^ VN{G x G),~~

~~• The fundamental operator W E x G)) is defined through~~

~~Wf{x, y) xy) (/ G L\G xG),x,yeG)~~

• Let the right regular representation oi G p2 ： G ——>• p2(工)(0 ：二 € *

~~知-1 G L^(G))) be defined via: ("2�K))(?/ ):= (C e e~~

~~• For any unit vector ^ G define~~

~~• Define V e through~~

~~G L^G))~~

~~Proposition 4.3.1. Let G be a locally compact group. Then~~

~~(a) W is a unitary and~~

~~W^fix.y) = W-'fix,y) 二 f(x’x-iy) (x,y e C)~~

~~(b) For any unit vector ^ G m^ e B(G x G)~~

~~(c) V is self-adjoint and V*X2{X)V = P2{X) {XEG)~~

~~(d) (V* 0 ii)LHG)W* = 汤~~

~~(e) (P2(工）0 ^{p2{x) 0 10^2(0)) for all x e G~~

~~(f) 工)）二 ~~il>LHG)W = A2W ® A2(a;) for all xeG Amenability of Certain Banach Algebras 53~~~~

~~~~Proof, (a): f ( [ m{x,y)\'dx)dy JG JG 二 JG /(/ JG \i(x,xy)\^dx)dy= JG JG [ ([ \ax,xy)\'dy)dx=JG JG [ (/ \ax.y)\'dx)dy implies that W is 8L unitary.~~~~

~~~~Therefore~~~~

~~~~(b) : Let :GxG ^{L^G)), (gji) ^ 入 2(g)P2(h). Then tt,,,^ G Egxcv^ and thus h)=< 兀入2’^2(仏C > is an element in B{G x G) for all unit vector ^ e L^{G).~~~~

~~~~(c): For any 4,77 eL2(G),~~~~

~~~~=fc A{x)^^{x)r}{x-^)clmc{x-')~~~~

~~~~It follows that V*'rjix) = Vrf(X-^)A(X)—2 = T]{X) For the second identity, let ( e L'^{G),x,y G G. Then~~~~

~~~~=A{x)^^7]{yx)) = {p2{x)r]){y)~~~~

~~~~Therefore,, V*X2(x)V = p2{x). (d): For any ^ G L^G x G), x.yeG,~~~~

~~~~={V* Anieiiability of Certain Banach Algebras 54~~~~

~~~~二神)�Or-i’y)) =iDi.2(G)胁’?/))~~~~

~~~~(e): Let xeG, f e L^G x G). Then~~~~

~~~~二义之0：))~~~~

~~~~= A{xy/^f{y,yzx)~~~~

~~~~=P2{x))f{y,yz)~~~~

~~~~=(^^L^G)^ p2{x))Wf{y,z)~~~~

~~~~(f): Let 0； eCr.fe X G). Then~~~~

~~~~[r*{X2{xmf{y,z))~~~~

~~~~So, r*(A2(.T)) == A2(:r) % A2(a;) [x e G) •~~~~

~~~~Proposition 4.3.2. With above notations, is a Hopf-von Neumann~~~~

~~~~algebra and hence it induces a natural completely contractive Banach algebra structure~~~~

~~~~on A{G). Moreover, the multiplication。厂^^…。’）(机st the usual pointwise product ofA{G).~~~~

~~~~Proof. We simply write T* := for convenience.~~~~

~~~~Clearly, T* is a unital, iiijective, normal, *-homomorphism.~~~~

~~~~Let fijo e A(G),x e G. Then~~~~

~~~~< /i Or* /2,A(a;) > =~~~~

~~~~==< fl ®/2，入2�(8)A2(:C) >~~~~

~~~~=< fuHx) > < f2^2(00) >~~~~

~~~~= fl-f2(x-')~~~~

~~~~=< fl ' fH^) >~~~~

~~~~Since X-ziG) is dense in VN{G), or* is just the pointwise product on A{G). Therefore I"'"" is CO-associative, and hence a comultiplication of VN{G). •~~~~

~~~~Theorem 4.3.3. Let G, H be two locally compact groups. Then~~~~

~~~~A{G X H) A(G)IA{H)~~~~

~~~~Proof. Refer to [Riia2] [Section 16.2]. •~~~~

~~~~For complex functions u, v on G, write uQv the function on G x G which is given by~~~~

~~~~(uQv)(x,y) := u{x)v{ij) {x,y G G)~~~~

~~~~Proposition 4.3.4. Let G be a locally compact group. Then we have:~~~~

~~~~八 (a) The A(G)-himodule actions of A{G x G) induced by the identification A{G)§)A{G)=~~~~

~~~~A{G X G) is given by~~~~

~~~~if • 9)ix,y) = f{x)g{x,y) (/ G ^(C), g e A{G x G), x.yeG)~~~~

~~~~and~~~~

~~~~(g. f){X.Y) 二 g(工,Y)f{y) (/ G A(G), g EA{GXG),x,yE G)~~~~

~~~~Hence, these A(G)-bimodule actions of A{G x G) extend to B{G x G) in a canonical~~~~

~~~~fashion.~~~~

~~~~(b) Define A : B(G x G) > B(G), {Af){x) := f{x,x) (x e G). Then it extends the~~~~

~~~~diagonal operator A : A{G x G) ^ ^ A{G). Anieiiability of Certain Banach Algebras 56~~~~

~~~~(a): Note that the natural ^(G)-bimodule actions on A{G) % A(G) is given by~~~~

~~~~(/i ^f2)-g = h® 129 and g . � /2) = 9fi ® /2 {fi,f2,g e A{G))~~~~

~~~~and~~~~

~~~~9-{fiQf2) = 9fiQf2 (fij2.geA{G))~~~~

~~~~Clearly, these actions are continuous and extends to the actions on A(G x G) stated in~~~~

~~~~the theorem. Therefore, it extends to the actions on B(G x G) via. the same formulae.~~~~

~~~~In addition, for any f G A{G),g e B{G x G),~~~~

~~~~\\f • 9\\b{GxG) < \\f O IhiGxcMlBiCxG) < ||/|丨鄉)丨丨"|丨寧xG)~~~~

~~~~and~~~~

~~~~\\g- fhiGxG) < WghicxoWf QM\B{GXG) < ||.9|U(gxg)||/|U(G)~~~~

~~~~The results follows.~~~~

~~~~(b): Note that the corresponding mapping induced by the isometric isomorphism i :~~~~

~~~~A(G)^A{G) ^ A{G X G) is given by~~~~

~~~~A ：華)O •—鋼，A(/ 0 g){x) = (fQ g�[x, x) = f(x)gix)~~~~

~~~~Define A : A{G x G) ^ = h{x,x). then~~~~

~~~~A(/i G /2))W = Mx)f2{x) = A{f, O f2)(x) (/i,/2 e A{G),x e G)~~~~

~~~~Since A and A agree on the dense subset A{G) © A{G), they are equal on A{G x G). • Anieiiability of Certain Banach Algebras 57~~~~

~~~~Lemma 4.3.5. Let G be an amenable locally compact group, and suppose that there is a~~~~

~~~~bounded net {7na)aeA in B[G x G) such that~~~~

~~~~and~~~~

~~~~ll/Am 厂0 (/6聊~~~~

~~~~Then A(G) is operator amenable.~~~~

~~~~Proof. Let (e以)爬丑 be a bounded approximate identity oi A{G) such that stzp/j^s 11^/3II — 1.~~~~

~~~~Since A{G x G) is a closed ideal of B(GxG), the net (e/3-ma.ef3)(a’f3)€AxB lies in A{GxG).~~~~

~~~~For any f G A{G),~~~~

~~~~f • (e/3 • rric • ep) - (e^ . m�. ep) • f]{x, y)~~~~

~~~~=f[x)ei3{x)mc{x,y)ef3{y) - e0(x)ma{x,y)e0{:y)f(y)~~~~

~~~~=e(3{x)e(i{y)[f{x)ma{x,y) - ma{x,y)f{y)~~~~

~~~~So,~~~~

~~~~f{x)A{ei3 • rria . ep){x) - f{x) == - f(x) implies that • m� 6/3•) - /IU(G)~~~~

~~~~=IL4/AM« — FLUIG) < ||E/3INI/||||AM. - /|U(G) + HF — /IU(G) - 0 •~~~~

~~~~Lemma 4.3.6. Let G be a locally compact group, and suppose that there is a net ((Jaevi of unit vectors in L'^(G) satisfying~~~~

~~~~0 (7] e L'iCr)) Anieiiability of Certain Banach Algebras 58~~~~

~~~~and~~~~

~~~~IIA2 � — 0~~~~

~~~~umformly on all compact subsets of G. Then the net (m《丄 in B(G x G) satisfies the~~~~

~~~~hypotheses of lemma 4.3.5.~~~~

~~~~Proof. Let f G A{G). By the polarization identity, we may suppose that~~~~

~~~~f{x) =< X2(x)ti,v > {x e G)~~~~

~~~~for some 7; G Note that for any x,y e G,~~~~

~~~~=< X2{x)r},7] >< > =< \2(x)rj,r] >< X2ix)V*X2iy)V^a,^a > by lemma 4.3.1(c)~~~~

~~~~and~~~~

~~~~(jn“ •/)(工•，~~~~

~~~~=< X2(x)p(yKcy,^a >< >~~~~

~~~~=< X2{x)V*X2(y)V^a,^a >< > by lemma 4.3.1(c)~~~~

~~~~=< (A2(X) 0 idL^(G)Yy"\2{y)V (8) A2(7/))(^a <8) 乂。③〉~~~~

~~~~=< (A2(a:) (8> �idLHG))My) ® 入2(?/))(乂 (S) id^XC^ 汤”>~~~~

~~~~=< (A2(X) (g) idmoW* ® idL2^G))W*{\2(y) 0 ® .似 L2(G))0 rj),�� > ” by lemma 4.3.1(f)~~~~

~~~~=< (A2(x) 0 idL2(G)W(y* ® i(kHG))(My) O idL^G))iy ® % ”仏 ®il> by lemma. 4.3.1(c), (d)~~~~

~~~~=< W*{X2(x) 0 id^^oWiV* (8)~~~~

~~~~=< (A2(:c) 0 \2{x)W\2{y)V O ® ")，(8) 77)� Anieiiability of Certain Banach Algebras 59~~~~

~~~~Then we obtain~~~~

~~~~if . rn� -m� ./)(工,y) =< {\2{x)V* X2{y)V ® H^Ma _ — (g) ” 仏 0'0>~~~~

~~~~+ < {X2(x)V''X2{y)V 0 rj), ® 77) >~~~~

~~~~Therefore,~~~~

~~~~11/•爪— M。• /IIB(GXG) < 2|M|||~~~~

~~~~Since (a; 入之⑷“，">:rj e Cc(G)} = Cc(G) * Cc(G)" is dense in A{G), we may~~~~

~~~~suppose that~~~~

~~~~f{x) =< X2(x)r},r]> {x e G)~~~~

~~~~for some r/ E Cc(G).~~~~

~~~~Note that for any x e G, we have~~~~

~~~~(/Am,J(a;)~~~~

~~~~=/(咖“工，工） =< >< >~~~~

~~~~=< >< � >by lemma 4.3.1(c)~~~~

~~~~=< (M^) V*A2(a:)l/)&(g)~~~~

~~~~=< (g) ® by lemma, 4.3.1(f)~~~~

~~~~=< W*{\2{x) 0 idL2^c)){idL2{G) % V*X2(x)V)W'{^a by lemma, 4.3.1(e)~~~~

~~~~二< VR 场 ⑷ O A2(aO)(ic/巧G) ® VWda ③")，③>~~~~

~~~~=< ® V*)W*iX2{x) (g) idL2^G))W(idL2^G) ® by lemma 4.3.1(f)~~~~

~~~~=< (A2(a；)� ® (8) W{idj,2^c)� �”）>~~~~

~~~~It follows that~~~~

~~~~(/Am�-/)�如(魄>~~~~

~~~~+ < (8) id^^oYia (8) ")’ W{i.dmG) ® VWi^a Anieiiability of Certain Banach Algebras 60~~~~

~~~~Therefore, we have~~~~

~~~~LL/AKJ — /II < 2\\RJ\\\\W{IDI^2^G) ^ VOH^FC^ ® V)"II~~~~

~~~~Let K ：二 supp". Since ||入2�P2� - — 0 uniformly on /(，we obtain:~~~~

~~~~JG JG~~~~

~~~~=[f - rj(x)Uy)\'dydx JG JG =F I"�|A(:r)—洲2办办 J K JG =F \ri{x)NX2{x)P2{x)^a-U\ldx-^0 J K So, ||/AKJ-/||^0. •~~~~

~~~~Theorem 4.3.7. Let G be a locally compact group, and let A{G) be its Fourier algebra.~~~~

~~~~Then the following statements are equivalent:~~~~

~~~~(a) G is amenable.~~~~

~~~~(b) A{G) is operator amenable.~~~~

~~~~Proof, "(a) (b)" See [Rual] or [Run] [Chapter 7'.~~~~

~~~~"(b) (a)” By theorem 4.1.19, A(G) has a bounded approximate identity. Therefore,~~~~

~~~~G is amenable by theorem 3.2.2. •~~~~

~~~~4.4 Ideals in A{G) with bounded approximate identi-~~~~

~~~~ties~~~~

~~~~We shall see in this section that the amenability of G can be characterized by the existence of bounded approximate identity in certain ideals of A[G)\ G is amenable if and only if~~~~

~~~~/(/f)(defined below)~~~~

~~~~G. Anieiiability of Certain Banach Algebras 61~~~~

~~~~Definition 4.4.1. Let G be a locally compact group.~~~~

~~~~(a) For any E C G, a. closed ideal I{E) of A{G) is defined by~~~~

~~~~1(E) := {u e A(G) : = 0 for all x e E}~~~~

~~~~(b) For any closed ideal I < define~~~~

~~~~I丄{T e VN{G) :< T’u>=0 for each u e 1}~~~~

~~~~(c) For any subgroup H of G, define VNH(G) to be the von Neumann algebra of VN{G)~~~~

~~~~generated by {入2� ^• € i/}.~~~~

~~~~Lemma 4.4.2. Let G be a locally compact group and H a closed subgroup of G. For any~~~~

~~~~V E A(H), there exists u G A{G) such that U\h = v and ||叫二 ‘Zn/{|i?"i||A(G):…// — t'}~~~~

~~~~Proof. Refer to [Her] [Theorem 1]. •~~~~

~~~~Theorem 4.4.3. Let G he a locally compact group. Then we have:~~~~

~~~~(a) For any closed ideal I < ^{G), f 丄 is an A{G)-submodule of VN{G).~~~~

~~~~(b) For each closed subgroup H of G, A{G)/I{H) is isometrically isomorphic to A(H).~~~~

~~~~Hence, I[H)丄 is isometrically isomorphic to VNH(G).~~~~

~~~~(c) VNH{G) is isomorphic to VN(H).~~~~

~~~~Proof. Refer to [For2] [Lemma 3.8] and [F-W] [Proposition 4.2, 4.3]. •~~~~

~~~~Lemma 4.4.4. Let H be an amenable locally compact group. Then there exists a completely contractive projection P : > VN{H).~~~~

Proof. Note that VN(H) is the commutant of {p2(aO ： x e H}. Let m be a left invariant mean on L°°{H). For convenience, we regard m as a finitely additive measure on H. Now define P : £(L^(//)) to be the weak operator converging integral

~~~~P(T)= [ p2{x)Tp2{xydmH{x) JH Anieiiahiiity of Certain Baiia.ch Algebras 62~~~~

~~~~for each T € That is,~~~~

~~~~= [ dmH(x) JH for each f,g E L'^(H). From the invariance of m, it is easy to see that for each T G~~~~

~~~~P2(X)P{T) = P(T)P2(X)~~~~

~~~~for all X e H. It follows that P{T) G VN{H). It is also clear that if T G VN(H), then~~~~

~~~~P{T) e VN(H).~~~~

~~~~We have that P is completely positive, since each amplification P„, : ~^~~~~

~~~~M.n{VN{H)) is given by the weak operator converging integral~~~~

PnilTij]) = f diag{p2ix)) [T,,] diag{p2{x)ydmH{x) JH for each [T”.] G M„(£(L-(//))), where diag{p2{x)) denotes the nxn diagonal matrix with all diagonal entries equal to p2(x). Finally, since P{I) —• I, P is completely contractive. n

~~~~Proposition 4.4.5. Let G he a locally compact group and let H be an amenable dosed~~~~

~~~~.subgroup of G. Then there exists a completely contractive projection from VN{G) onto~~~~

~~~~/(//)丄.~~~~

~~~~Proof. By lemma 4.4.4’ there exists a completely contractive projection P : ~~>~~~~

~~~~VN(H). let : VN(H) ~^ VNH{G) C be a (isomorphism. By the Arveson-~~~~

~~~~Wittstock extension theorem,~~~~

~~~~: VNH{G) VN(H) has a completely contractive extension~~~~

~~~~Since 少（/) = /,少 is completely positive by lemma, 4.1.7.~~~~

~~~~Let P $。尸。: Z{L\G)) 一 VNH{G) - /(//)丄 of Certain Banach Algebras 63~~~~

~~~~Then Q P\VN{G) is the required projection. •~~~~

~~~~Theorem 4.4.6. Let G he a locally compact group. Then the following statements are~~~~

~~~~equivalent:~~~~

~~~~(a) G is amenable.~~~~

~~~~(b) I{H) has a hounded approximate identity for any closed proper subgroup H of G.~~~~

~~~~(c) /(//) has a hounded approximate identity for some closed amenable subgroup H of~~~~

~~~~(d) 1(H) has a hounded approximate identity for some closed proper subgroup H of G.~~~~

~~~~Proof.~~~~

~~~~"(a) (b)” Suppose that G is amenable, then A(G) is operator amenable. For any closed~~~~

~~~~subgroup H, H is amenable by theorem 2.1.6. By proposition 4.4.5, I{H) is completely~~~~

~~~~weakly complemented in the operator amenable completely contractive Banach algebra~~~~

~~~~A{G). Therefore, I{H) has a bounded approximate identity by theorem 4.1.21.~~~~

~~~~"(b) (c)”’ "(b) (d)" Trivial.~~~~

~~~~“(c) (a)" Assume that H is an amenable closed subgroup of G such that I{H) has a bounded approximate identity. Theorem 3.2.2 and theorem 4.4.3 imply that A{G) 11(H)=~~~~

~~~~A{H) has a bounded approximate identity. If A{G)/I{H) and I{H) both have bounded approximate identity, then it is easy to construct a bounded approximate identity for~~~~

~~~~“(d) (a)" Since H is proper, there exists x e G \ H. Furthermore I{xH) has a bounded approximate identity {ua}aeA- Assume that v 6 Cc{H) n 丑)with suppv ==~~~~

K C H. We can find a neighborhood V of K in G and a n G B(G) such that u(K) = 1, suppw. C V and VDXH = 0. By lemma 4.4.2, there exists Ui e A{G) such that U]\H = v. Anieiiability of Certain Banach Algebras 64

~~~~Let w 二 uiu. Then w e I{xH) and t/^// 二 v. let Va = Ualn- Then Va G A(H)~~~~

~~~~and < ||wa|U(G) by lemma 4.4.2. Therefore, for any v e Cc{H) fl and~~~~

~~~~w G I{xH) with '⑴I// = V,~~~~

~~~~0 < — v\\a(H) < limjiv"； — W|U(g) 二 0~~~~

~~~~Since A(II)nCc(II) is dense in A(H), {va}a^A is a bounded approximate identity in A{H).~~~~

~~~~It follows that H is amenable. Therefore, I{H) has a bounded approximate identity for~~~~

~~~~some closed amenable subgroup H of G. By “ (c) (a)", G is amenable. •~~~~

~~~~Remark. Lemma 4.4.4 and lemma 4.4.5 are [F-K-L-S] [lemma 1.1] and [Proposition 1.2~~~~

~~~~respectively. Theorem 4.4.6 is a mixture of [F-K-L-S] [Corollary 1.6], [For2] [Theorem 3.9]~~~~

~~~~and [For3] [Theorem 3.9].~~~~

~~~~4.5 Other Cohomological properties of A{G)~~~~

~~~~As we shall see in this section, other evidences for the suitability of the theory of operator spaces for the study of Fourier algebras are: A(G) is always operator weakly amenable~~~~

~~~~(compare with theorem 2.1.9) and is operator biprojective if and only if G is discrete~~~~

~~~~(compare with theorem 2.1.12).~~~~

For a completely contractive Banach algebra A, let MA ： ~> A,a ® b ^ a • b be the left module multiplication map of the ^"-bimodule A, and let TTIA ： A^^A^——> A' ^ be the multiplication map. Denote:

~~~~(a) /(A Ker{MA)~~~~

~~~~(b) [K] A] := span {a - u - u • a : u e K, a e A}~~~~

~~~~(c) A? := KerirriA)~~~~

~~~~(d) K^ := Ki n~~~~

~~~~Remark. It is not hard to see that K^ := K^ n (� Anieiiability of Certain Banach Algebras 65~~~~

~~~~Definition 4.5.1. A completely contractive Banach algebra A is said to be operator~~~~

~~~~weakly amenable if every completely bounded derivation D : A——> A* is inner.~~~~

~~~~Theorem 4.5.2. Let A be a commutative completely contractive Banach algebra. Then~~~~

~~~~A is operator weakly amenable if and only 二乂 and (KQY = (A§)A). Kf.~~~~

~~~~Proof. Refer to [Spr] [Theorem 2.2]. •~~~~

~~~~Let G be a locally compact group and let n 6 ^q. Denote by 1 : G ——C the constant~~~~

~~~~function or the trivial representation of G, put A.,^ := span{x H->< 7r(x), 7] >: 77 G //兀}'' ""(口）~~~~

~~~~and write VNj^ the von Neumann algebra generated b)' 7r(G) in £,{H7r).~~~~

~~~~Definition 4.5.3. Let G be a locally compact group and let tt, cr G ^Q.~~~~

~~~~(a) Let TY X a : G X G ~> U(J~L节 x Hp) be the Kronecker product of TT and cr, given by~~~~

~~~~(tt X a){s,t) = 7r{s)^a(t) {s,t e G)~~~~

~~~~(b) TT and a are said to be disjoint if they do not have subrepresentations which are~~~~

~~~~unitary equivalent.~~~~

~~~~Theorem 4.5.4. Let G be a locally compact group and let ir, a E YLg- Then we have:~~~~

~~~~(a) =~~~~

~~~~(b) VN^mN, = VKxo~~~~

~~~~(c) If TT and a are disjoint, then ViV_� =VN^ 田⑷ VN�and A^^^ = Ar ®i .4^.~~~~

~~~~Proof. Refer to [Spr] [Introduction of Section 2]. •~~~~

~~~~Theorem 4.5.5. Let G be a locally compact group. Then the representations 1x1，A2 x 1,~~~~

~~~~1 X X2, A2 X A2 are all disjoint and we have the following complete isometric isomorphism:~~~~

~~~~(a) AiG)^ ^ ^A.ei~~~~

~~~~(b) A{Gy§A{G) ^ ^(A2xA2)e(ixA2) = span{u, Ixv.ue A(G x G) and v e A{G)] Anieiiability of Certain Banach Algebras 66~~~~

~~~~Proof. Refer to [Spr] [Prop 3.1，3.2]. •~~~~

~~~~Theorem 4.5.6. Let G be a locally compact group and let /1(G) be its Fourier algebra.~~~~

~~~~Then A[G) is always operator weakly amenable.~~~~

~~~~Proof. If G is compact, then A{G) is operator amenable and hence operator weakly~~~~

~~~~amenable.~~~~

~~~~subalgebra of B(G x G). Write KI = KF^^^ and KQ = KQ^^I~~~~

~~~~In the identification A(Gy§A(G)^ = A^,, the multiplication map uia^g) corresponds to~~~~

~~~~the map~~~~

~~~~R ： A^ ~~AX2®I,RU(S) = u(s, s) (s e G)~~~~

~~~~Let GD •= {(5, s) : s e G} be the diagonal subgroup of G x G. It is easily seen that~~~~

~~~~R{u) — U\GD- Then~~~~

~~~~Ki = Ker(R) = span{u - 1 x R(u):u - R{u) x 1 : u G AJ and~~~~

~~~~KO = Ker{R) n x G) = I{GD)~~~~

~~~~Note that I[GD) is a set of spectral synthesis.(See [Her]) We thus obtain that~~~~

~~~~Since A{G x G) is a closed ideal of we get~~~~

~~~~{A{G)^A(G)) . KI = A{G x G) . Ker{R) C A{G xG)N Ker{R) = I{GD) = KO~~~~

~~~~On the other hand, by (*)，~~~~

~~~~KO = I(GD) = AIG X G) • HGD) C A{G X G) • Ker{R) ^ (A{G)®A{G)). K, Anieiiability of Certain Banach Algebras 67~~~~

~~~~We thus have~~~~

~~~~• K, = KO =荷~~~~

~~~~By the Taiiberian theorem for A{G), A{GY = Hence, A{G) is operator weakly~~~~

~~~~amenable by theorem 4.5.2. •~~~~

~~~~Remark. Above lemmas and theorems can be founded in ppr .~~~~

~~~~The following theorem can be considered as a "dual" result of theorem 2.1.12 and its~~~~

~~~~proof can be founded in [Ad .~~~~

~~~~Lemma 4.5.7. Let A be a commutative, operator biprojective, completely contractive~~~~

~~~~Banach algebra. Then cr(A), the spectrum of A, is discrete.~~~~

~~~~Proof. Refer to [Ari] [Theorem 7.29，7.30]. •~~~~

~~~~Theorem 4.5.8. Let G be a locally compact group. Then G is discrete if and only if~~~~

~~~~A(G) is operator biprojective.~~~~

~~~~Proof. Assume that G is discrete. Recall that the diagonal operator induced on A{G x G)~~~~

~~~~is given by A : ^(G' x G) A(G), A{f{x)) := f{x, x) for any x e G. For f e A{G).~~~~

~~~~denote by p{f) the function on G x G such that~~~~

~~~~Pif)(x,x) = fix, x) {xeCr)~~~~

~~~~and~~~~

~~~~pUXt, y) = 0 (x, y ec�xi^ y)~~~~

~~~~Let f{x) 二< 77 > for some ^,77 € P{G) and let ^ 二 EXGG ^^XX- and - Wy where iVy)yeG € Then~~~~

~~~~fKfX 工,y) =< ® X2{x)^\ rf > where�‘=an d rf = T.yecVyXy,y Moreover, = and~~~~

~~~~= ！whence l|/?(/)|L4(axG) < 丨|/||刷.Consequently, p{f) e A{G x G). AnieimbiJity of Certain Banach Algebras 68~~~~

~~~~It is easily seen from the definition that p : A(G) ~> A(G x G) is a homomorphism which is a right inverse of A. Conversely, since o{A{G)) = G, the result follows by lemma 4.5.7. •~~~~

~~~~An alternative proof of this theorem can be found in [Woo2 . Appendix A~~~~

~~~~Objects or notions and their ” dual” versions~~~~

~~~~OBJECTS AND NOTIONS "DUAL" OBJECTS OR NOTIONS~~~~

~~~~—s. G G~~~~

~~~~L\G)~~~~

~~~~M{G) B(G) C^ C；^~~~~

~~~~L卞） VN{G)~~~~

~~~~UCB(G) UCB{G)~~~~

~~~~II • LU II • LIE-~~~~

~~~~II • LIE- II • lloo~~~~

~~~~Discrete group Compact group~~~~

~~~~Compact group Discrete group~~~~

~~~~69 Appendix B~~~~

~~~~Results and ''dual results,’ for general locally compact groups~~~~

~~~~RESULTS "DUAL" RESULTS L\G)~~~~

~~~~Li(Gi) ^ L1(G2) if and only if Gi ^ G2 A{Gi) - A(G2) if and only if Gi ^ G2~~~~

~~~~M(Gi) = M(G2) if and only if Gi ^ G2 B{Gi) ^ B{G2) if and only if Gi ^ G2~~~~

~~~~乙 1(G)* ^ ^(G)* ^ VN{G)~~~~

~~~~CoiGY ^ M{G)~~~~

~~~~L\G) is II • ||c.(G)-dense in C*{G) A{G) is || • ||oo-dense in Co{G)~~~~

~~~~Z/i(G) is amenable if and only if G is amenable A{G) is operator-amenable if and only if G is amenable~~~~

~~~~L^{G) is weakly amenable A{G) is operator weakly amenable~~~~

~~~~L^{G) is biprojective if and only if G is compact A{G) is operator biprojective if and only if G is discrete~~~~

~~~~L1(G) is unital G is discrete A{G) is iinital G is compact ^ L\G)'^Ma{G) = M{G) 台 A{G) = B{G)~~~~

~~~~LHGi) ®L\G2) = X G2) A(Gi) |a(G2)臣c_i. A(G] x G2)~~~~

~~~~70 Appendix C~~~~

~~~~Results and ” dual results” which are equivalent to the amenability of the group~~~~

~~~~—Sult厂 "DUAL RESULTS" WHICH ARE EQUIVALENT~~~~

~~~~TO THE AMENABILITY OF THE GROUP~~~~

~~~~乙 1(G) has a bounded approximate identity A{G) has a bounded approximate identity~~~~

~~~~C*{G) 二 LUG) * C*(G) Co(G) = A{G) • Co(G)~~~~

~~~~V{G) is closed in {G), L\G)) A{G) is closed in m{A{G),A{G))~~~~

~~~~L^{G) factors(weakly) factors (weakly)~~~~

~~~~Li(G) is (7(Afi:G),Co(G))-clense in M{G) A{G) is g(B(G)’C7*(G))-dense in B{G)~~~~

~~~~m{L\GlL\G))^M{G) m{A{G\A{G))^B{G)~~~~

~~~~UCB{G) = O(G) * L°°(G) * U (G) A(G). VN(G)~~~~

~~~~is a closed subalgebra of L°°(G) is a closed subalgebra of VN(G)~~~~

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